Constructible complexes on algebraic varieties

AI-generated exposition by GPT-6.1 Sol and GPT-6 Astra (OpenAI), October 2026. Public domain (CC0).

A point of a singular space has two different kinds of local data. A stalk records what a sheaf assigns at the point. A costalk also records how that value attaches to the surrounding punctured space. We will calculate that attachment in three examples, then show how finite stratifications and duality organize the answers. The puncture contributes a monodromy operator, a node contributes a diagonal map between branch values, and a cone contributes the cochains of a circle bundle.

The classical setting is a complex algebraic variety with its classical topology and coefficients in a characteristic-zero field \(\Lambda\). The arithmetic setting is a separated finite-type scheme over a finite field or its algebraic closure, with coefficients \(\overline{\mathbb Q}_\ell\), where \(\ell\) is invertible on the scheme. Finite torsion coefficients will be used to construct the arithmetic operations; the passage to normalized adic coefficients is treated explicitly in Section 3. A local system has finite rank, and an arithmetic local system means a lisse sheaf.

Our convention is \(H^m(K[a])=H^{m+a}(K)\). In particular, a vector space \(V[a]\) lies in degree \(-a\). The Tate coefficient \(\Lambda(1)\) has geometric Frobenius \(q^{-n}\) at an \(\mathbb F_{q^n}\)-point.

1. Compute the attaching map first

Let \(i:\{x\}\hookrightarrow X\) and \(j:X\setminus\{x\}\hookrightarrow X\). For a complex \(K\), name its point value and boundary value

Restriction gives a map \(a:A\to B\). The supported restriction is its derived fibre:

To see the map as well as the degrees, take an injective resolution. Sections supported on the closed set form the kernel of restriction to the open complement. Appendix B.2 constructs the resulting support triangle, and B.3 takes its point restriction to obtain (1.1). Appendix A.1 proves the exactness of stalk colimits and the resolution properties used here. Thus a punctured-neighbourhood computation must include its restriction maps and the attaching map; a list of cohomology dimensions alone does not determine the costalk.

Proposition 1.1. The boundary stalk

For \(j:\mathbb C^\times\hookrightarrow\mathbb C\), a local system with fibre \(V\) and positive monodromy \(T\) satisfies

Proof. Appendix A.4 constructs transport, proves the winding-number classification by explicit angle lifts, and constructs the local system with prescribed monodromy as a quotient of the constant sheaf on the logarithmic covering. It proves that morphisms are exactly the intertwiners of the monodromy operators. We use those constructions with the fixed fibre \(V\) and positive-loop operator \(T\).

Cut a punctured disc along each of two different rays. The two resulting contractible angular regions overlap in two components. Trivialize along one overlap; transport on the other supplies \(T\). The two-cover section differential is

Put \(u=y-x\). Subtracting the first target coordinate from the second changes this differential to \((u,y)\mapsto(u,(T-1)y)\). Its first summand is an acyclic identity complex; the second is the asserted complex. A.3 proves acyclicity of the angular regions, and A.4 proves the two-cover comparison with derived sheaf sections, so the calculation is a derived one. Shrinking the disc preserves the cover and its matrices under radial transport. Taking the exact colimit over these discs therefore computes the actual boundary stalk. Its groups are \(\ker(T-1)\) in degree zero and \(\operatorname{coker}(T-1)\) in degree one. \(\square\)

Proposition 1.2. Stalks and costalks

Writing \(B=[V\xrightarrow{T-1}V]_{[0,1]}\), the extensions of \(L\) have the following boundary data:

Proof. On an open direct image, the adjunction to restriction is the identity after restricting to that open; the injective model of B.2 gives the identity boundary map and zero supported kernel. Extension by zero has zero point stalk. Their fibres in (1.1) give the first two rows. In the last row \(T=1\). The fibre is represented in degrees \(0,1,2\) by \(\Lambda\xrightarrow1\Lambda\xrightarrow0\Lambda\); cancelling the identity gives the answer. Thus \(j_!L\) has costalk groups \(\ker(T-1)\) and \(\operatorname{coker}(T-1)\) in degrees one and two. \(\square\)

For the étale constant sheaf on \(\mathbb G_m\subset\mathbb A^1\), the boundary groups are \(\Lambda\) in degree zero and \(\Lambda(-1)\) in degree one; smooth purity places \(\Lambda(-1)\) in the line's degree-two costalk. Section 3 gives the finite-coefficient purity argument; Appendices BV–BX treat its coefficient comparisons. A general arithmetic local system requires the full inertia action. The single-operator description applies to tame local monodromy; it does not remove wild inertia.

Proposition 1.3. A node separates two branch values

At an ordinary node \(p\) on a reduced complex algebraic curve, let \(N=\nu_*\Lambda_{\widetilde X}\) for the normalization. Then

All unlisted costalk groups are zero.

Proof. Appendix L derives the two smooth normalization branches and the local analytic equation \(uv=0\) from the completed-local node criterion. A small neighbourhood is two discs joined at their centres, with puncture the disjoint union of two punctured discs. By Proposition 1.1 and the compatible branch computation in B.4, its boundary value is \([\Lambda^2\xrightarrow0\Lambda^2]_{[0,1]}\). A constant value at the node restricts to the same value on each branch, so the attaching map is the diagonal. Formula (1.1) gives

The diagonal is injective; its cokernel is one-dimensional. For \(N\), the two centre values are independent and the attaching map in degree zero is the identity of \(\Lambda^2\). Cancelling this identity leaves degree two alone. B.4 identifies the supported complex with local cohomology by its section resolution, and proves the branch description of \(N\). No identification of a costalk with an ordinary stalk has been made. \(\square\)

Taking the difference of the two branch values gives the exact sequence

This is local at the chosen node; globally there is one quotient summand for each node. For the cubic \(y^2z=x^2(x+z)\), B.5 constructs the normalization as \(\mathbb P^1\), identifies the two points over \([0:0:1]\) as \(t=1,-1\), and checks that every other point is smooth. Consequently (1.2) is global for this particular cubic. At a smooth point both \(\Lambda_X\) and \(N\) have stalk \(\Lambda\) and degree-two costalk \(\Lambda\).

Over an algebraically closed field, the finite-coefficient étale calculation uses the same stalkwise sequence and purity on the normalization branches. Its costalk groups are \(\Lambda\) in degree one and \(\Lambda(-1)^2\) in degree two. Over a finite field these statements describe geometric vector spaces: Frobenius may permute the two branches, acting also on \(\Lambda^2/\Delta\Lambda\). They assert trivial branch action only for a split node. The passage to adic coefficients uses the operations and coefficient comparisons of Appendices BV–BX.

Proposition 1.4. The circle bundle around a plane-curve cone

Let \(C\subset\mathbb P^2(\mathbb C)\) be smooth of degree \(n\geq1\), with genus \(g\), and let \(X\) be its affine cone. The link \(M=X\cap S^5_\epsilon\) is the oriented circle bundle of \(\mathcal O_C(-1)\), whose Euler class is \(-n\eta\), where \(\int_C\eta=1\). With our characteristic-zero field,

Proof. A nonzero point of the cone is a line of \(C\) together with a nonzero vector on it. Normalizing the norm of that vector identifies the link with the unit vectors of the tautological line. C.3 verifies these maps, the cone topology, and the comparison of compatible punctured-neighbourhood sections with link cochains.

E.1–E.4 construct the connected compact curve and its finite oriented triangle decomposition. They give integral cohomology \(\mathbb Z,\mathbb Z^{2g},\mathbb Z\) in degrees \(0,1,2\). Counting a hyperplane section gives degree \(n\) for \(\mathcal O_C(1)\), hence Euler class \(-n\eta\) for its dual. The integral Thom construction in Appendix D yields

Over \(\Lambda\), the displayed map from degree zero to degree two is invertible. Exactness therefore identifies \(H^1(M)\) with \(H^1(C)\) and \(H^2(M)\) with another copy of \(H^1(C)\). The endpoints give \(H^0(M)=H^3(M)=\Lambda\), and the other terms vanish. E.5–E.6 prove the plane-curve genus formula by the tangent/normal sequences and a ramification count; specifically \(\omega_C\simeq\mathcal O_C(n-3)\), \(\chi(C)=3n-n^2\), and \(\deg\omega_C=n(n-3)=2g-2\). These supply the numerical value used above. \(\square\)

Proposition 1.5. Recover the vertex costalk from the link

For the cone vertex \(v\), the only nonzero costalk groups of the constant sheaf are

Proof. The attaching map is \(\Lambda[0]\to C^\bullet(M;\Lambda)\), sending a scalar to its constant cochain. C.3 proves this statement for the actual shrinking cone neighbourhoods, using the cochain/sheaf comparison in C.2. Its fibre has \(H^k=\widetilde H^{k-1}(M;\Lambda)\). Connectedness removes the reduced degree-zero term; (1.3) gives precisely (1.4). For \(j:X\setminus\{v\}\hookrightarrow X\), the boundary value of \(Rj_*\Lambda\) is \(C^\bullet(M;\Lambda)\), with identity attaching map and zero costalk. \(\square\)

For a smooth plane cubic, \(g=1\). Thus the link dimensions are \(1,2,2,1\), the costalk dimensions are \(2,2,1\) in degrees \(2,3,4\), and \(i_v^*Rj_*\Lambda[2]\) lies in degrees \(-2,-1,0,1\). This last calculation is the input for the later intermediate-extension example.

The algebraic punctured cone is also \(\mathcal O_C(-1)^\times\). In the étale setting its global Gysin calculation gives \(H^1(C)\), \(H^1(C)(-1)\), and \(\Lambda(-2)\) in positive degrees \(1,2,3\), with \(\Lambda\) in degree zero. This global computation uses the étale Gysin formalism; it does not identify a strict-local vertex costalk. The proof of (1.4) is the classical cone argument just given.

2. Assemble local data and dualize it

A finite algebraic stratification partitions \(X\) into smooth locally closed subvarieties, with each closure a union of pieces. A sheaf is constructible when its restrictions to the pieces of some such partition are finite-rank local systems. For a fixed partition \(\mathcal S\), let \(D^b_{\mathcal S}(X)\) consist of bounded complexes whose cohomology sheaves have this property. Allowing the partition to vary gives \(D^b_c(X)\). The condition is on cohomology sheaves; resolution terms need not be constructible.

Proposition 2.1. Closure under triangles and truncation

The category \(D^b_{\mathcal S}(X)\) is triangulated and stable under ordinary cohomological truncation. The same holds for \(D^b_c(X)\).

Proof. Kernels and cokernels of maps of local systems are local systems: on a common trivializing neighbourhood the map is a constant matrix. An extension is locally constant as well. Classically, shrink to a convex coordinate neighbourhood. The vanishing of \(H^1\) of the constant kernel, proved in A.3, lifts a basis of the constant quotient to sections of the middle sheaf and splits the extension there. Appendix N constructs the finite-monodromy correspondence and its strict lisse coefficient passage. In the global-lattice lisse category, kernels and cokernels correspond to the invariant kernel and quotient representations. Identifying extensions taken in the full constructible category with lisse extensions uses the normalized adic comparison specified in Section 3.

For a triangle \(A\to B\to C\to A[1]\), the long cohomology sequence expresses each \(H^m(C)\) as an extension of a cokernel by a kernel of maps between the cohomology of \(A,B\). Restricting to a stratum applies the preceding argument. Boundedness is preserved, and ordinary truncations retain a subset of the same cohomology sheaves. For finitely many constructible objects, take a common finite smooth refinement; K.6 proves the classical algebraic refinements used here. The same triangle argument then proves the assertion for \(D^b_c\). \(\square\)

Proposition 2.2. Finite assembly from strata

Every object of \(D^b_c(X)\) is obtained by finitely many triangles from shifts of local systems extended by zero from strata.

Proof. Choose a constructibility stratification and remove its open strata, with inclusions \(j:U\hookrightarrow X\), \(i:Z\hookrightarrow X\). The exact extension/restriction sequence of B.2 gives

The closed term has fewer strata. Repeat until a single stratum is left. On that stratum, ordinary truncation triangles assemble the bounded restriction from its cohomology local systems. Both inductions are finite. This produces triangles, without asserting that their extension maps vanish. \(\square\)

For the structural map \(a:X\to\mathrm{pt}\), define

Appendix F constructs the classical structural right adjoint and identifies its local pairing with the dual of compactly supported sections. H constructs the relative operations and their pre-bidual exchanges. I–J establish the normal-neighbourhood geometry and finite local cohomology needed to prove constructible biduality. K then verifies the algebraic constructibility hypotheses for general separated finite-type morphisms. Consequently, in the classical setting,

These are natural maps with the evaluation and adjunction counits described in those appendices. Applying internal Hom to a shift also gives \(D(K[a])=(DK)[-a]\). For a geometric closed point the two restriction formulas are

F.6 proves the second formula from the closed-embedding adjunction; J.2 and J.5 prove the first using finite evaluation and biduality. M.5–M.7 give the separate singular finite-coefficient étale argument over fields. The arithmetic formulas use the finite-coefficient constructions of M and the complete-coefficient constructions of BV–BX, with their stated geometric hypotheses. Formula (2.1) concerns geometric closed points; a generic-point restriction has a dimension correction.

Proposition 2.3. Smooth orientation fixes the shift

For a smooth complex variety of pure dimension \(d\) and finite-rank local system \(L\),

The arithmetic counterpart is \(\omega_X\simeq\Lambda_X(d)[2d]\), \(D_XL\simeq L^\vee(d)[2d]\), subject to the coefficient foundations of Section 3.

Proof. C.4–C.5 compute the compact cohomology of a real \(m\)-ball and the sign of its orientation transitions. The local compact-support dual defining \(a^!\Lambda\) is therefore \(\mathrm{or}_X[m]\), as proved in F.5. A complex coordinate change has positive real determinant, so the orientation sheaf is canonically constant and \(m=2d\). Locally \(L\) is finite free; internal Hom gives \(L^\vee\otimes\omega_X\), and these identifications agree under its transition matrices.

For finite étale coefficients, Proposition 3.3 constructs the trace and proves the smooth formula with its Tate twist. The coefficient dual is exact over the self-injective ring \(\mathbb Z/n\), also for finite stalks that are not free. The derived-complete and rational forms need the additional normalized adic comparison; no single finite reduction proves them. \(\square\)

In particular, \(D(L[d])=L^\vee[d]\) classically and \(L^\vee(d)[d]\) arithmetically. Both the shift and the twist survive the calculation.

Proposition 2.4. Constructibility of the dual

For a classical algebraic Whitney stratification \(\mathcal S\), duality preserves \(D^b_{\mathcal S}(X)\). For finite étale coefficients, a fixed constructibility system must also be compatible with boundary direct images of the dual local systems. Without that compatibility the assertion allows refinement.

Proof. For the given classical Whitney stratification, Appendix I constructs compatible normal products. J.1–J.3 prove the coefficient comparison and finite link cohomology. In a normal product the support and attaching complexes are constant along a small stratum box; J.4–J.5 combine them with its orientation to compute the dual there. Its bounded cohomology is consequently finite-rank and locally constant on that stratum. K.6 supplies a finite algebraic Whitney refinement when needed.

For finite étale coefficients, Proposition 2.2 reduces the assertion to \(b_!L\) for a stratum inclusion. The pre-bidual identity (M.12) gives \(D_Xb_!L=Rb_*D_SL\). Smooth duality, finite direct-image constructibility in M.4, and the stipulated compatibility make this constructible for the chosen system. M.5 supplies the field descent. The finite triangle construction finishes the argument. The complete and rational coefficient forms use the operation and duality comparisons of BV–BX at their stated hypotheses. The compatibility issue is also explained in the free BBD text, §2.2.10. \(\square\)

We can now dualize the concrete attaching calculations. For a puncture, G.3–G.4 prove

The dual transport operator is \((T^{-1})^\vee\). For the node, the complex orientations on the two normalization branches identify \(D_XN=N[2]\); G.4 verifies agreement on overlaps and under permutation of branches. Dualizing (1.2) therefore gives the actual triangle

Thus the node's dual constant sheaf has stalk \(\Lambda^2\) in degree \(-2\) and \(\Lambda\) in degree \(-1\), with costalk \(\Lambda\) in degree zero. The triangle includes the attaching maps and does not specify a canonical splitting. The corresponding geometric étale triangle has \(N(1)[2]\); branch permutations must still be retained over a finite field.

For the cone, G.2–G.3 verify finite link cohomology with local coefficients and both dual exchanges for the actual radial model. Applying (2.1) to (1.4) places the dual constant sheaf's stalk groups in degrees \(-2,-3,-4\), with dimensions \(2g,2g,1\); its costalk is \(\Lambda\) in degree zero.

Proposition 2.5. Euler functions discard extension data

The function \(\chi_K(x)=\sum_m(-1)^m\dim H^m(i_x^*K)\) is integer-valued and constructible. For every triangle \(A\to B\to C\to A[1]\), it satisfies \(\chi_B=\chi_A+\chi_C\).

Proof. Each stratum has constant cohomology ranks, and the sum is finite by boundedness. Stalk restriction preserves a triangle. In its finite long exact cohomology sequence, write the dimension of each term as the dimension of its incoming image plus that of its outgoing image. The alternating contributions of those images cancel, giving the asserted identity point by point. \(\square\)

For a puncture, \(T-1:V\to V\) has kernel and cokernel of equal dimension. Hence every \(Rj_*L\) has boundary Euler value zero, though its two boundary groups may be large. For the node, (1.2) says that the normalization's Euler function exceeds the constant function by the indicator of the node. These two examples show exactly what the Euler function retains from the attaching calculation.

3. Operations and the coefficient passage

For a separated finite-type algebraic morphism \(f:X\to Y\), the six-operation assertion is preservation of bounded constructibility by \(Rf_*\), \(Rf_!\), \(f^*\), \(f^!\), derived tensor product and internal \(R\mathcal Hom\). It allows compatible refinements of source and target stratifications. Appendices H–K supply the classical construction and proof. The finite-coefficient étale arguments are given below and in Appendices M and BF–BJ. Appendices BK–BX give the complete and rational coefficient comparisons at their stated hypotheses.

The adjunctions to be used are

In a Cartesian square, with \(g:Y'\to Y\), \(f':X'\to Y'\), and \(g':X'\to X\), the proper-support comparisons are

For proper \(f\), \(Rf_!=Rf_*\), giving arbitrary base change for ordinary direct image in that case. Étale smooth base change gives another ordinary-image comparison when \(g\) is smooth and torsion coefficients are prime to the characteristic. For an open immersion, \(j^!=j^*\) and \(j_!\) is exact. Ordinary direct images of general analytic maps need not be algebraically constructible; K.8 uses algebraicity in its compactification and stratification argument.

Here are the exact written inputs for the arithmetic construction. The proper base change theorem, Theorem 1.1, gives the proper-comparison argument for arbitrary torsion sheaves, with bounded-below comparison in Proposition 11.2 and the geometric-fibre formula in Corollary 11.3. The proper-comparison foundations and reductions used here are proved in AU–BE. AU.1–AU.6 prove the Noetherian Chow modification, proper coherent finiteness, formal functions, Stein factorization and the proper finite-fibre consequence used in those foundations. AV.1–AV.6 supply arbitrary-base approximation, the separated finite-type ambient construction, general Chow modification and Stein factorization. AW.1–AW.6 prove separated quasi-finite factorization over a qcqs base and the finite, finitely presented refinement of an étale covering. AX.1–AX.7 prove henselian-pair idempotent lifting, the degree-zero restriction theorem for arbitrary sheaves, finite direct image over arbitrary bases and full faithfulness for finite étale covers. AY.1–AY.6 prove faithfully flat section descent, geometric field-extension invariance, the nonaffine strict-local section formula and general proper degree-zero base change. AZ.1–AZ.6 prove constructible-coefficient compactness, finite-stage descent and embeddings into finite direct images over arbitrary qcqs bases. BA.1–BA.7 prove affine closed-subscheme effacement for arbitrary torsion coefficients, including affine étale lifting, finite-cover vanishing and arbitrary-base higher continuity. BB.1–BB.4 prove henselization along an arbitrary ideal, affine henselian comparison in all degrees and the restricted-injective bound, including its coefficient-category justification. BC.1–BC.6 prove coherent algebraization, finite-cover and torsor lifting over complete Noetherian pairs and the complete-local projective-line comparison. BD.1–BD.2 prove finite-coefficient descent for finite étale covers and torsors over arbitrary qcqs affine-transition limits. BD.3–BD.6 prove pointed approximation on local integer models, finite-cover and torsor lifting for proper schemes over arbitrary henselian local rings, and projective-line cohomology comparison there. The proper schemes need not be finitely presented over the henselian base. BE.1–BE.8 supply the residue-characteristic curve comparison, general proper base change for arbitrary torsion sheaves and arbitrary base changes, bounded-below comparison, and the geometric-fibre formula. Smooth base change and local acyclicity gives the smooth-comparison argument for quasi-compact, quasi-separated direct image, including affine inverse limits of smooth base changes, with torsion stalk orders invertible on the base. Its local-acyclicity assertion for a smooth map requires locally constant coefficients. AL.1–AL.3 prove its finite-map and curve-invariance inputs. AM.1–AM.6 prove its connected-fibre neighbourhood construction over arbitrary bases, using the discrete valuation witness and integer-model descent proved in AM.4. AN.1 proves degree-zero comparison with this same generality. AN.2–AN.6 supply the completed-section and punctured formal-neighbourhood arguments. AO.1–AO.6 prove the coherent cover-algebra extension, flat-base-change compatibility and relative purity over Noetherian bases. AP.1–AP.7 prove regular local purity and the tame root-base-change calculation for the Galois DVR extensions used here. AQ.1–AQ.5 prove codimension-one gluing and descent at the split section, completing the generic-cover theorem over a DVR. AR.1–AR.5 prove the extension over an arbitrary integral normal Noetherian base, including non-quasi-compact smooth families. AG.5 proves finite normalization of integer models in finite function-field extensions. AS.1–AS.7 prove the positive-degree cohomological induction and bounded-below comparison over arbitrary bases for each invertible coefficient modulus and the stated filtered unions, including affine limits of smooth changes of base. AT.1–AT.5 prove the resulting strict-local acyclicity, selected-neighbourhood colimit and ordinary smooth-locus nearby-cycle comparison. Neither result treats an adic complex as a torsion sheaf.

The pro-étale site and ℓ-adic complexes constructs coherent derived reductions and completion. Its constructibility criterion uses bounded constructible reduction on a quasi-compact Noetherian scheme, with uniform bounds supplied by completeness. Appendix N proves the finite-cover monodromy equivalence, effective affine descent and strict lisse coefficient correspondence used in this passage, including stable lattices and the finite congruence cover. A torsion sheaf's derived reduction may occupy two degrees; ordinary quotients lose that information. ℓ-adic sheaves and their cohomology treats strict-system normalization, finite coefficient models, rationalization and compatible perfect complexes, retaining separate geometric finiteness and scalar-extension inputs. Appendix BK constructs the actual two-term reduction complexes, their transition maps, the completion unit, and its compatibility with direct image and base-change adjunctions. It also proves the classical/pro-étale comparison used at each finite level. These coefficient constructions do not themselves give constructibility or enlarge the geometric scope of smooth base change.

Proposition 3.1. Smooth comparison after completion

Let \(\mathcal O\) be a complete discrete valuation ring of residue characteristic \(\ell\), invertible on \(Y\). Let \(f:X\to Y\) be quasi-compact and quasi-separated, and let \(g:Y'\to Y\) be smooth, or an affine inverse limit of smooth changes of base. Let \(K\) be derived complete with bounded-below finite reductions coming from the étale site. Write \(g^*_{\mathrm{comp}}\) for pullback followed by derived completion. Then the canonical map

is an isomorphism.

Proof. Work with the constant coefficient sheaf \(\underline{\mathcal O}\) on the pro-étale site, and write \(C\) for derived completion. This is not an identification of the constant sheaf with its completed coefficient sheaf. Put \(K_n=K\otimes^L_{\mathcal O}\mathcal O/\pi^n\). BK.3 represents this by \([\mathcal O\xrightarrow{\pi^n}\mathcal O]\otimes K\), in degrees \(-1,0\), with transition components \((\pi,1)\). BK.4 constructs \(C(K)=\operatorname{holim}_nK_n\) and proves that its unit induces an isomorphism on every derived reduction. Its proof supplies the actual cone maps and needs no uniform bound on the reductions.

BK.5 proves \(Rf_*K_n=(Rf_*K)_n\) by those finite free terms and their multiplication map. Since direct image is a right adjoint, it preserves the homotopy limit; hence \(Rf_*C=C Rf_*\) with its unit compatibility. Define (3.1) by applying \(C\) to the ordinary adjunction map \(g^*Rf_*K\to Rf'_*g'^*K\) and then using this last comparison on the target. BK.6 identifies its two objects and its arrow as \[ \operatorname{holim}_n g^*Rf_*K_n \xrightarrow{\operatorname{holim}_n\beta_{K_n}} \operatorname{holim}_n Rf'_*g'^*K_n. \] The identification follows from the \(\mathcal O\)-linear adjunction units and counits applied to the specified two-term complexes. It respects transition maps and successive changes of base.

BK.1–BK.2 prove the site and coefficient comparisons needed to identify each \(\beta_{K_n}\) with its bounded-below étale torsion comparison. A power of \(\ell\) kills \(\mathcal O/\pi^n\), so the required torsion orders are invertible on the base. AS.1–AS.7 prove this smooth comparison over arbitrary bases for each such invertible modulus, including affine limits of smooth base changes. Each level map is then an isomorphism, and so is the displayed homotopy limit. The lower bound may vary with \(n\). The coefficient proof uses a uniformizer and its two-term resolutions, so it applies to every complete discrete valuation ring in the statement, without requiring a finite residue field or a finite extension of \(\mathbf Z_\ell\).

The general-base geometric proof uses the connected-neighbourhood and degree-zero arguments AM–AN, the normal Noetherian generic-cover extension AR, and the arithmetic-model normalization AG.5 inside AS.4–AS.6. Thus each required finite-level comparison is established at the stated qcqs and smooth-limit scope, and the preceding completion argument proves (3.1). No finiteness or bounded constructibility of \(Rf_*K\) is deduced from this comparison. The constructibility and normalized adic constructions are treated separately in BM–BX. \(\square\)

Proposition 3.2. Finite coefficients and proper supports

Let \(k\) be a field, let \(\ell\) be invertible in \(k\), and let \(\Lambda\) be a finite ring killed by a power of \(\ell\). Let \(f:X\to Y\) be a separated morphism of finite type between separated finite-type \(k\)-schemes. If \(K\) is bounded with constructible \(\Lambda\)-cohomology, then \(Rf_!K\) is bounded constructible. For a constructible sheaf \(F\),

is finite and vanishes for \(q>2\dim X_{\bar y}\). In particular, compactly supported cohomology over a separably closed field is finite; for proper \(X\), ordinary cohomology is finite too. If the fibre dimensions are at most \(d\) and \(K\) has cohomology in \([a,b]\), then \(Rf_!K\) has cohomology in \([a,b+2d]\).

In this finite-coefficient statement, constructibility means finite local constancy on a finite partition into locally closed subschemes. Smoothness of these pieces over \(k\) is not required, so the statement includes imperfect fields. An empty fibre has zero cohomology.

Proof. We use the constructed proper-support functor, its composition, arbitrary base change, excision and Mayer–Vietoris from Cohomology with compact support. Its proper cohomological-dimension proof gives the bound \(2\dim X_{\bar y}\) before a general finiteness theorem is used: compactify the fibre and extend by zero. Consequently all the spectral sequences below have finite contributing ranges. That companion's general constructibility assertion is not an input to this proof.

First take \(\mathbf F_\ell\)-sheaves. Appendix P proves the curve input needed here: compactly supported cohomology of a constructible prime-to-characteristic sheaf on a separated curve over an algebraically closed field is finite and vanishes above degree two. It uses the programme construction of the curve's Picard scheme, proves finiteness and surjectivity of invertible multiplication on its points by infinitesimal lifting, and then gives the Kummer, finite-boundary and trace arguments. No formula for the degree of multiplication or the rank of Jacobian torsion is required for this deduction. AK.1–AK.5 prove the continuity, exact finite-curve pushforward and ordinary/compact finiteness assertions at the smooth-curve scope used in the pairing argument. The stronger curve computations in Torsion sheaves on curves retain their own stated scope. Over a separably closed field one may first pass to its algebraic closure, a purely inseparable extension which does not change the étale topos.

We will need proper smooth families of curves without assuming general constructibility. If \(h:C\to S\) is such a family, with \(S\) locally Noetherian, proper base change and the preceding curve theorem make every geometric stalk of \(R^qh_*\mathbf F_\ell\) finite. Every specialization map is an isomorphism by Smooth base change and local acyclicity, Corollary 11.3. That result uses smooth local acyclicity and proper base change, without a constructibility hypothesis.

Here is why these two facts imply finite local constancy. For any étale \(\mathbf F_\ell\)-sheaf \(A\) with finite stalks and invertible specialization maps, choose a basis of \(A_{\bar s}\). Lift its finitely many elements to sections on an affine Noetherian étale neighbourhood \(U\) of \(\bar s\). They give \(\varphi:\underline{\mathbf F_\ell^r}\to A|_U\), an isomorphism at \(\bar s\). Naturality and the invertible specialization maps of both sheaves make the isomorphism locus stable under specialization and generization. Geometric lifts of these relations exist in strict localizations; changing a lift conjugates the maps and preserves invertibility. Delete the finitely many irreducible components of \(U\) not containing the distinguished point. Every remaining generic point is a generization of that point, so \(\varphi\) is an isomorphism there. Every other point specializes from one of these generic points, so \(\varphi\) is an isomorphism on all geometric stalks of the remaining neighbourhood. It is an isomorphism of sheaves. Thus \(R^qh_*\mathbf F_\ell\) is finite locally constant. No general direct-image constructibility theorem has entered this deduction.

A separated quasi-finite morphism has exact proper-support image and preserves constructibility. Indeed, Zariski's Main Theorem factors it as an open immersion followed by a finite morphism. Pushforward, pullback and finite morphisms proves exact finite direct image and étale topological invariance. Constructible sheaves and extension by zero proves constructibility of extension by zero and finite direct image from finite presentations and generic finite-algebra splitting. These elementary results precede the theorem we are proving.

We next prove that \(Rf_!\mathbf F_\ell\) is constructible for a smooth affine family of curves \(f:X\to S\), by induction on \(\dim S\), working on an affine base. In dimension zero, its finitely many geometric stalks are finite by the curve calculation, and the étale topos is that of finitely many residue fields. A continuous action on a finite stalk is trivialized by a finite separable extension, which proves constructibility in this case.

For the induction step it suffices to prove the assertion near each generic point of \(S\); the closed complement has smaller dimension, and arbitrary proper-support base change permits induction there. The geometric curve-completion input has the following exact form. After shrinking to a neighbourhood \(U\) of a generic point and making a finite surjective radicial change \(V\to U\), there is a finite map

Here \(T_i\to V\) is finite étale, \(C_i\to T_i\) is smooth proper with geometrically connected curve fibres, and \(Y_i\subset C_i\) is open. There is an open \(W\subset X_V\), dense in every fibre, for which \(\nu^{-1}W\to W\) is a nilpotent thickening; each \(\nu^{-1}W\cap Y_i\) is dense in its curve fibres. Appendix O constructs this diagram for the smooth affine family used here, with the stronger conclusions \(\coprod_iY_i=X_V\), \(\nu=\mathrm{id}\), \(W=X_V\), and finite boundary \(\coprod_i(C_i\setminus Y_i)\to V\). Zero-dimensional smooth generic components become finite étale after shrinking and need no curve argument. The construction supplies the geometric step in the finiteness argument of [AI Integrated Stacks].

The complement \(X_V\setminus W\) is quasi-finite over \(V\), since all its fibres have dimension zero. The complement of \(\nu^{-1}W\) in \(\coprod_i C_i\) is quasi-finite for the same reason. Étale topological invariance identifies \(W\)'s constant-sheaf image with that of \(\nu^{-1}W\). Apply excision first to \(W\subset X_V\) and then to \(\nu^{-1}W\subset\coprod_i C_i\). The two boundary terms are constructible by the quasi-finite case. The proper middle term is

which is constructible by the proper smooth curve result and exact finite direct image. The two triangles prove the assertion over \(V\). The finite surjective radicial map is a universal homeomorphism, so topological invariance descends it to \(U\). Induction completes the constant-coefficient case. A finite affine cover and the proper-support Mayer–Vietoris triangle extend this result to smooth separated families of curves whose source need not be affine; all affine intersections remain smooth families of curves.

Now let \(L\) be finite locally constant on an open \(U\subset\mathbf A^1_S\). On a connected component its monodromy has finite image \(G\), by N.2–N.3. We construct the subgroup needed for the reduction. Write \(|G|=\ell^a m\), with \(\ell\nmid m\), and let \(G\) act by left multiplication on its \(\ell^a\)-element subsets. Their number is prime to \(\ell\): the coefficient of \(t^{\ell^a}\) in \[ (1+t)^{\ell^a m}=(1+t^{\ell^a})^m \quad\text{in }\mathbf F_\ell[t] \] is \(m\). This identity follows by iterating the binomial formula in characteristic \(\ell\). At least one orbit therefore has size prime to \(\ell\). If \(P\) stabilizes a subset in that orbit, the orbit is in bijection with \(G/P\), so \(\ell^a\) divides \(|P|\). But left multiplication by \(P\) acts freely on that subset, partitioning its \(\ell^a\) elements into sets of size \(|P|\). Thus \(|P|=\ell^a\), as required. This proves the Sylow existence assertion used here.

Let \(g:U'\to U\) be the finite étale cover corresponding to the transitive \(G\)-set \(G/P\), constructed in N.3. Its degree \(e=[G:P]\) is prime to \(\ell\). Pulling back this cover gives a diagonal section, so \(\pi_1(U')\) fixes the chosen coset and the pulled-back representation factors through \(P\). On a nonzero finite \(\mathbf F_\ell\)-vector space, every nontrivial \(P\)-orbit has size divisible by \(\ell\). The number of fixed vectors is consequently divisible by \(\ell\); it includes zero, hence is at least \(\ell\). Choose a nonzero fixed vector and apply the same argument to the quotient by its line. Induction on dimension gives a filtration with trivial one-dimensional quotients.

Étale locally the cover is a disjoint union of \(e\) copies of the base. The diagonal map into those copies and their sum are invariant under permuting them, so they glue to unit and trace maps. Their composite is multiplication by \(e\) on every stalk. Thus

Multiplying the trace by \(e^{-1}\) splits the unit. The smooth separated curve-family case, the filtration, composition of proper-support images and this retraction prove constructibility of \(R(U\to S)_!L\). There are only finitely many connected components, so the result holds without connectedness.

For arbitrary constructible \(F\) on \(\mathbf A^1_S\), induct again on \(\dim S\). Choose an open \(U\) on which \(F\) is locally constant and which contains all generic points of the fibres over the generic points of \(S\). After shrinking \(S\) to a dense open, its complement \(Z\) is quasi-finite over \(S\). To justify the shrinking, the non-quasi-finite locus of \(Z\to S\) is closed in \(Z\); its image is constructible and contains no generic point of \(S\). Its closure therefore misses those generic points and can be deleted. In the excision triangle

the outer terms are constructible by the lisse and quasi-finite cases. The closed complement of the chosen dense base open is handled by induction and base change. This proves the result for the affine-line projection.

Finally work on an affine open of \(Y\), and take a finite affine cover of \(X\). Its intersections are affine because \(f\) is separated. Mayer–Vietoris reduces to an affine source. Embed that source as a closed subscheme of \(\mathbf A^n_Y\). Closed direct image is exact and constructible. Factor the projection of affine space into \(n\) affine-line projections, and apply the result just proved at each stage. The finite cohomological-dimension bound permits the ordinary cohomology spectral sequence, or finitely many truncation triangles, to pass from sheaves to bounded constructible complexes. This proves the assertion for \(\mathbf F_\ell\).

For a finite \(\ell\)-primary \(\Lambda\)-sheaf, filter its underlying abelian sheaf by powers of \(\ell\). The finitely many successive quotients are constructible \(\mathbf F_\ell\)-sheaves. Exact sequences and the preceding result give finite constructible underlying cohomology sheaves of its proper-support image. They are constructible as \(\Lambda\)-sheaves too: on each finite locally constant piece, the actions of the finitely many elements of \(\Lambda\) are morphisms of local systems. Bounded truncation again handles \(K\). Formula (3.2) and the degree bound were established by proper-support base change and proper cohomological dimension. For a structural map to a separably closed field, a constructible output has finite groups; if the map is proper, \(Rf_!=Rf_*\). The amplitude bound follows from the same finite spectral sequence. \(\square\)

For \(\mathcal O\) the valuation ring of a finite extension of \(\mathbf Q_\ell\), this proposition applies to every \(\mathcal O/\pi^m\). It proves the finite-level proper finiteness used in the nearby-cycle argument. BM constructs proper support on complete complexes with bounded-below classical reductions, proves its choice-independent composition and arbitrary base change, and proves bounded constructibility defined by first derived reduction. BN proves uniform finite-level Tor bounds, closure under completed tensor and inverse image, and the completed proper-support projection formula with its coherent comparison maps. BO proves the finite-coefficient local finite-free theorem and constructs the actual compatible adic local model on each connected stratum. BP proves the equivalence of strict constructible systems with complete constructible sheaves in degree zero, including torsion and all their sheaf morphisms. BQ proves normalized constructible cohomology, bounded standard truncations, the strict-system heart, and t-exact completed inverse image and open extension. BR proves equivalence with the specified derived category of cartesian pro-étale coefficient diagrams, including all higher morphisms and their limit sequence. BS proves the corresponding étale-diagram comparison, including transitions and higher morphisms. BT proves bounded-derived realization at each finite coefficient ring, constructs the strict adic heart realization functor, and proves its low-degree comparisons and exact higher-effacement criterion. BU proves full adic heart realization in dimension zero over finite or algebraically closed fields, including torsion coefficients. BV constructs the adic exceptional right adjoint on whole coefficient diagrams, proves ordinary-image and internal-Hom constructibility, and proves singular biduality with its operation exchanges. BW proves the equivalence with actual completed-ring module complexes and constructs central rationalization, its heart and operations. BX proves the full comparison with constructible rational sheaves, including higher morphisms, integral models and ordinary-operation comparisons. BY proves integral and rational heart realization on curves. BZ–CB prove the finite-coefficient geometric fibre comparison (BY.26), including all geometric base points and the characteristic-two descent. BY.7 deduces higher-dimensional realization over finite or algebraically closed fields, allowing a finite extension of a finite ground field and using the coefficient hypotheses of BO–BY throughout. The proposition does not assert nonproper ordinary-direct-image finiteness.

Proposition 3.3. Smooth trace from the affine line

Let \(\Lambda=\mathbf Z/n\), with \(n\geq2\) invertible in \(k\), and let \(f:X\to S\) be separated and smooth of relative dimension \(d\), between finite-type \(k\)-schemes. For every coefficient complex \(A\), there is a canonical isomorphism

Its counit gives the smooth trace \(Rf_!\Lambda(d)[2d]\to\Lambda\), compatible with base change and composition. This is the finite-coefficient formula; adic normalization is an additional coefficient passage.

Proof. BF.1–BF.6 construct the locally small unbounded derived categories of presheaves and sheaves, their triangles and coproducts, and exact derived sheafification with a fully faithful right adjoint. The resolution existence theorem used there is proved for general Grothendieck categories in Sheaves of modules and their derived categories, Theorem 7.2. The explicit cone identities in B.1 supply the triangulated structure, without a boundedness restriction. BF.5 verifies the compact-generator hypotheses for the earlier programme representability theorem, and BF.6 deduces representability on the sheaf category. Hence every exact coproduct-preserving functor out of this category has an exact right adjoint. No bound on the cohomological dimension of the field is used.

BG.1–BG.6 construct the exact, coproduct-preserving proper-support functor on arbitrary ordinary complexes at the present field scope. The finite relative cohomological bound proves unbounded proper comparison and preservation of sums; the compactification and proper–open exchange arguments give independence and composition. Apply BF.6 to \(H_B(K)=\operatorname{Hom}(Rf_!K,B)\). The representing objects define \(f^!B\); uniqueness of representing maps makes this a functor and gives the adjunction counit. Composition identifies right adjoints of composite maps. Étale upper shriek is ordinary inverse image, since étale lower shriek is its exact left adjoint. The resulting local Hom test is written in Poincaré duality for smooth varieties, Section 8, with the complete arbitrary-complex local and internal adjunction in BI.1. BH.1–BH.4 construct the unbounded tensor product on the actual étale site, its inverse-image compatibility and internal-Hom adjunction. BH.5–BH.7 prove the proper-support projection formula and its canonical comparisons, including the invertible-twist adjunction used below. We now construct the trace from the affine line.

For \(p:\mathbf P^1_S\to S\), the unit and the Kummer class of \(\mathcal O(1)\) give a map

It is an isomorphism on every geometric stalk by proper base change and the curve calculation: the class has degree \(+1\) and generates the degree-two group. It is therefore an isomorphism. Restriction to the infinity section is the identity on the first summand and zero on the second, because \(\mathcal O(1)\) restricts there to a trivial line bundle. Excision identifies, canonically,

BI.2 gives the geometric-fibre calculation and specifies this isomorphism as the composite (BI.6). Thus \(Rq_!\Lambda(1)[2]\to\Lambda\) has a specified trace, compatible with arbitrary base change: both the unit and the degree-one Kummer class pull back to their counterparts. For an étale \(u:U\to\mathbf A^1_S\), define its relative curve trace by composing the étale counit with this trace. Étale refinement preserves it. On geometric fibres it is the curve trace, because Poincaré duality for curves proves compatibility of the degree-normalized trace with open extension and finite maps; factor the quasi-finite étale map into these two kinds.

We next justify \(q^!A=q^*A(1)[2]\) for arbitrary \(A\), without a higher-dimensional trace. Use the dimension-one local argument of Poincaré duality for smooth varieties. Its neighbourhoods of \(\mathbf A^1_S\) are precisely étale maps of the preceding form. AI.3–AI.4 give the uniform-refinement, derived-cone and descent details at this exact scope. Here are the geometric inputs and the resulting maps. AT.1–AT.4 prove that the filtered ordinary cohomology of the selected strict-local fibre is \(\Lambda\) in degree zero and zero in higher degrees, using the exact pointed pairs and selected base maps of (AT.9). The curve-duality argument, with its ramification and finite-counting inputs proved in AJ.1–AJ.4, identifies the finite compact groups with the duals of those ordinary groups. BI.3 proves that their actual étale trace transitions are the transposes of pullback. Killing finitely many ordinary generators makes the unwanted compact transition maps zero. In top degree the trace kernel is killed in the same way, while the trace is surjective.

Constructibility from Proposition 3.2 makes the zero-map loci of these finite sheaf systems open in the constructible topology. Compactness of that topology gives one refinement killing all stalk maps; the fixed cohomological range and the bounded-ghost lemma give a further refinement killing the derived cone map. The strict-local bases cause no change of hypotheses: every finite neighbourhood diagram descends to a finite-type étale base neighbourhood, and proper-support base change supplies its constructible sheaves on the strict base. AI.4 descends the zero cohomology maps by writing their constructible image supports as finitely many locally closed finite-presentation schemes, and then kills the derived maps by four successive ghost refinements.

BI.4 constructs the required zero cone map by a finite chain of triangle completions. It does not assume a functorial cone on the entire neighbourhood category. BI.5 then proves directly that precomposition by the trace induces an isomorphism of the filtered Hom groups in (AI.7), by lifting each morphism and killing its obstruction at a later neighbourhood. The two colimits are \(H^{r+2}(A(1))_{\bar s}\) and \(H^r(q^!A)_{\bar x}\). BI.1 and BI.5 identify this map with the stalk map adjoint to the specified trace and projection formula, for arbitrary unbounded \(A\). Its isomorphism on every stalk proves the affine-line formula. This uses the constructed affine-line trace, curve duality, the selected-neighbourhood calculation and Proposition 3.2; no higher-dimensional trace has been assumed.

Compose the affine-line traces to obtain the trace of an affine space. BJ.1 proves the exact top-cohomology composition and source-cover descent used for this construction. BJ.3 proves that the product trace is invariant under coordinate permutations: the geometric product class is a tensor of degree-two generators, whose exchanges have positive sign.

For an étale coordinate chart \(u:U\to\mathbf A^d_S\), compose its summation counit with this affine-space trace. BJ.4 proves independence of the chart directly on top cohomology. After geometric base change, the coordinate lemma of BJ.2 joins two charts, locally, by finitely many changes of one coordinate. Permute that coordinate to the last position. The common first \(d-1\) coordinates define a smooth relative curve, and its two traces agree by the relative-curve construction of BJ.3. The source-cover exact sequence and geometric stalks then give equality of the original top trace maps. This constructs a global, coordinate-independent \(t_f:R^{2d}f_!\Lambda(d)\to\Lambda\).

The bound \(Rf_!\Lambda(d)[2d]\in D^{\leq0}\) makes this top trace specify a unique derived map \(T_f:Rf_!\Lambda(d)[2d]\to\Lambda\), by (BJ.11). For arbitrary \(A\), transpose

On each coordinate chart this map is the composite of the affine-line isomorphisms and exact étale inverse image, because its counit is the same composed trace. BI.1's local adjunction and exact stalks therefore prove that its transpose is the isomorphism (3.7). BJ.5 gives this argument for unbounded \(A\), using BH's arbitrary-complex projection formula.

BJ.6 checks composition by the explicit interchange of étale summation with affine-space trace, and checks base change on the actual counit mate. In particular, for \(X_1=X\times_S\mathbf A^1_S\), with projections \(q_X:X_1\to X\), \(q:\mathbf A^1_S\to S\), and induced \(f_1:X_1\to\mathbf A^1_S\), it gives

For a parameter section \(s_t\), transitivity and the invertible-twist adjunction give \(s_t^!q^*B=B(-1)[-2]\). BJ.6 identifies its positive generator with the negative supported Kummer boundary \(-\lambda\kappa(t-t_0)\), whose trace is \(+1\) by AI.2 and BI.2. These parameter comparisons follow from the constructed coordinate-independent trace. Thus the trace, its smooth orientation and the base-change and composition maps are all specified. \(\square\)

If \(X\) is smooth of dimension \(d\) over a separably closed field, the result gives \(a^!\Lambda=\Lambda(d)[2d]\). If \(i:Z\hookrightarrow X\) is a closed immersion of smooth varieties of pure codimension \(c\), transitivity for their structural maps and the invertible twist gives

BJ.6 proves the same identity for arbitrary coefficients pulled back from a common smooth base and identifies its Gysin counit and composition. This proves purity for these smooth pairs. It does not assert the general absolute-purity theorem for arbitrary regular schemes or traits.

For a finite locally constant \(F\) on smooth \(X\) over an algebraically closed field, the internal-Hom adjunction proved in the same companion now gives

The coefficient ring \(\mathbf Z/n\) is self-injective, and finite-module evaluation is biduality, by the explicit extension and cardinality proof in M.6. Thus \(D_XF=F^\vee(d)[2d]\); Proposition 3.2 and the displayed comparison give finite mutually perfect pairings \(H^r_c(X,F)\times H^{2d-r}(X,F^\vee(d))\to\Lambda\), with evaluation followed by the constructed trace. This smooth local-system statement does not replace the full singular constructible Verdier-duality theorem.

The structural map \(a:X\to\mathrm{pt}\) gives \(Ra_*K=R\Gamma(X,K)\) and \(Ra_!K=R\Gamma_c(X,K)\). Proper-support base change identifies a geometric stalk of \(Rf_!K\) with compactly supported cohomology of the geometric fibre. For ordinary direct image the analogous fibre formula is asserted here when \(f\) is proper. These distinctions explain why the local computations in Section 1 retained their actual punctured-neighbourhood limits.

Appendix M extends the finite-coefficient construction over fields to ordinary direct-image finiteness, constructibility of the dual, singular biduality and the operation exchanges. It writes the finite-support and generic-restriction reductions explicitly. Its inputs include curve finiteness, cohomological continuity and the categorical exceptional adjunction identified in M.1.

The finite-coefficient operation statements occur in [SGA 4, XVI, Corollary 1.2] and [SGA 4, XVII, 5.2.6, 5.2.8, 5.2.9 and 5.3.6]. [SGA \(4\frac12\), Théorèmes de finitude, 1.1, 1.3 and 1.5–1.6] treat finiteness over a regular base of dimension at most one, including our fields; 4.3 requires finite Tor dimension for biduality, while 4.7 supplies the self-injective \(\mathbb Z/n\) variant. The field case is proved in Appendix M.

[Laszlo–Olsson, II] treats the adic theory: Theorem 3.20 and Remark 3.21 for bounded comparison and rational coefficients, Proposition 4.1 for internal Hom, Theorem 7.7 for duality, §§8–9 for operations, (9.6.1) for projection, and Theorem 12.1 with §12.3 for base change. BK–BN construct the completed operations, proper support, base change and projection specified there at the explicit reduction-defined scope used here. BO supplies the local finite-free and compatible adic model proofs. BP proves the strict-system comparison in degree zero, and BQ proves constructible cohomology, standard truncations and the resulting strict-system heart. BR proves the derived-cartesian pro-étale diagram comparison and its higher-morphism formula; BS identifies the corresponding whole étale diagrams and all their morphisms. BT supplies finite-coefficient realization and the adic heart realization functor, with degree-one comparison, degree-two injectivity and a necessary-and-sufficient effacement criterion. BU proves the full realization equivalence in dimension zero over finite or algebraically closed fields. BV constructs the adic exceptional adjunction, proves its coefficient compatibility, and establishes singular biduality and the dual exchanges. BW proves completed-ring comparison and central rationalization, including the torsion kernel and the conservative map to actual rational sheaf complexes. BX supplies the full rational-sheaf comparison, its higher morphisms and its operation comparisons. BY proves integral and rational heart realization on curves. BZ–CB prove its geometric fibre comparison (BY.26) for finite coefficient fields of characteristic invertible in the ground field. BY.7 gives higher-dimensional realization with the complete integral and rational coefficients specified in BO–BY, over finite or algebraically closed ground fields; the construction permits a finite extension in the finite-field case. The intended scope remains bounded constructible coefficients on finite-type schemes with the stated dimension bounds, including all the coefficient comparisons needed by the later lessons.

4. Exercises and complete solutions

Exercise 1 (easy). For \(j:\mathbb C^\times\hookrightarrow\mathbb C\), compute all stalks of \(Rj_*\Lambda\). Then compare the boundary stalks and costalks of \(Rj_*L\) and \(j_!L\) when \(T=J_2(1)\oplus(-1)\) on \(\Lambda^3\).

Solution. On the open stratum the constant direct image is \(\Lambda\) in degree zero. At zero, Proposition 1.1 has differential \(1-1=0\), so it gives \(\Lambda\) in both degrees zero and one and no other groups. For the specified rank-three system, choose a Jordan basis \(e_1,e_2,e_3\) with \((T-1)e_2=e_1\), \((T-1)e_1=0\), and \((T-1)e_3=-2e_3\). The kernel is spanned by \(e_1\); the cokernel is represented by \(e_2\). Thus the two boundary stalk groups of \(Rj_*L\) are one-dimensional in degrees zero and one. Its attaching map is the identity, so its costalk is zero. For \(j_!L\) the stalk is zero, and taking the fibre shifts those same boundary groups to costalk degrees one and two. All other boundary groups vanish. On the smooth open stratum both extensions restrict to \(L\); at any of its points their stalk is \(V\) in degree zero and their costalk is \(V\) in degree two, by the oriented disc calculation.

Exercise 2 (easy). Dualize \(j_!\Lambda\) on the punctured line. Specify the object, its boundary stalk and costalk, and the arithmetic twist.

Solution. The open-immersion exchange and the smooth orientation give \(D_{\mathbb C}j_!\Lambda=Rj_*\Lambda[2]\). Its stalk at zero is the complex \([\Lambda\xrightarrow0\Lambda]_{[-2,-1]}\), and its costalk is zero. This agrees with reversing and dualizing the two nonzero costalk degrees of \(j_!\Lambda\) in Proposition 1.2. Arithmetically the formula is \(Rj_*\Lambda(1)[2]\): the boundary groups are \(\Lambda(1)\) in degree \(-2\) and \(\Lambda\) in degree \(-1\). This arithmetic statement uses the purity and coefficient comparisons identified in Section 3; the shift alone does not account for Frobenius.

Exercise 3 (medium). Compute local cohomology at a node and compare it with normalization. Extend the calculation to an ordinary point with \(r\geq2\) smooth branches and distinct tangent lines.

Solution. L.5 identifies the actual separated smooth normalization branches, and B.4 proves the local section comparison. For \(r\) branches, the puncture has \(r\) circular components. The constant stalk-to-boundary map gives

Therefore \(H^1_{\{p\}}(X,\Lambda)=\Lambda^r/\Delta\Lambda\), of dimension \(r-1\), and \(H^2_{\{p\}}(X,\Lambda)=\Lambda^r\); the other local groups vanish. For normalization the first map is \(1:\Lambda^r\to\Lambda^r\), leaving just \(\Lambda^r\) in degree two. At a node \(r=2\), so the constant local groups have dimensions one and two, whereas the normalization has only the two-dimensional degree-two group. This difference is the branch-value quotient in (1.2). Finite-coefficient étale purity places \((-1)\) on the degree-two term; over a nonclosed field the branch-permutation representation is retained. The complete and rational coefficient formulations use the purity and finite-map comparisons of BV–BX at their stated hypotheses.

Exercise 4 (medium). Show that the stalk Euler function defines a homomorphism from the Grothendieck group of \(D^b_c(X)\) to integer-valued constructible functions. Compute it for the constant and normalization sheaves of a nodal curve.

Solution. The Grothendieck group is the free abelian group on object classes modulo the triangle relations \([B]=[A]+[C]\). Proposition 2.5 proves that the corresponding function relation holds at every point, so the assignment factors through that quotient. Each function is constructible because its cohomology ranks are constant on a finite stratification. The constant sheaf has function one. The normalization sheaf has function one on the smooth locus and two at each node: its stalk counts the separate branches. Their difference is the sum of node indicators, exactly as the stalkwise quotient sequence predicts. The triangle \(K\to0\to K[1]\) also proves \([K[1]]=-[K]\); the same sign follows by reindexing the Euler sum.

Exercise 5 (hard). Describe the link of the affine cone over a smooth degree-\(n\) plane curve. Compute its field and integral cohomology and its constant-sheaf vertex costalk; include the plane cubic.

Solution. A link point is a unit vector of the tautological line, so the link is the circle bundle of Euler class \(-n\eta\). Put \(b=(n-1)(n-2)=2g\). Over our field, the Gysin map \(\Lambda\xrightarrow{-n}\Lambda\) is invertible. The link dimensions in degrees \(0,1,2,3\) are consequently \(1,b,b,1\). The attaching fibre removes the constant degree-zero class and raises each remaining degree by one, so the vertex costalk dimensions in degrees \(2,3,4\) are \(b,b,1\). For a cubic \(b=2\).

Over \(\mathbb Z\), that map remains injective but has cokernel \(\mathbb Z/n\). The same exact sequence gives \(H^0(M;\mathbb Z)=H^3(M;\mathbb Z)=\mathbb Z\), \(H^1(M;\mathbb Z)=\mathbb Z^b\), and

Choose a lift of each free basis vector to split this sequence as abelian groups. The result is \(H^2(M;\mathbb Z)\simeq\mathbb Z^b\oplus\mathbb Z/n\), without a preferred splitting. For \(n=1\), the cone is a complex plane and its link is \(S^3\), the Hopf bundle over \(\mathbb P^1\). For \(n=2\), \(b=0\) but \(H^2(M;\mathbb Z)=\mathbb Z/2\). For \(n=3\), it is \(\mathbb Z^2\oplus\mathbb Z/3\). These checks also explain why characteristic zero was specified for the field formula.

Appendix A. Derived sections on the punctured disc

This appendix proves the classical sheaf-cohomology input of Proposition 1.1. Coefficients are vector spaces over a field \(\Lambda\). The construction concerns open subsets of Euclidean space and spaces homeomorphic to them. In particular, it applies to the angular coordinate regions used below. It does not establish the étale or adic comparisons in Section 3.

A.1. Resolutions and sections

Kernels of sheaves are computed on sections; cokernels are obtained by sheafifying the sectionwise cokernel. Both have the expected stalks. Indeed, equality or a lift of a germ is witnessed on a smaller neighbourhood. The same observation proves exactness of filtered colimits of vector spaces: a finite equation involving representatives can be tested at one common later index. Consequently stalks and filtered colimits commute with cohomology of complexes.

There are enough injective sheaves of \(\Lambda\)-vector spaces. For a point \(x\), let \(e_x\) denote its inclusion. The skyscraper \((e_x)_*V\) has sections \(V\) on an open containing \(x\), and zero otherwise. The identity

follows by taking germs. Its right side is exact in \(F\): stalks are exact, and a linear map from a subspace extends after extending a basis. Thus every skyscraper here is injective. A product of injectives is injective, since maps into it are products of maps into its factors and each component extension can be chosen separately. The germ map

is injective: a section with all zero germs is zero. Iterating (A.2) on the successive cokernels gives an injective resolution. We compute \(R\Gamma(X,F)\) by the sections of such a resolution. Two resolutions give the same derived object: extend a map degree by degree into the injective target, first on the image of the preceding differential; two extensions differ by a homotopy constructed by the same induction. The induction starts on the common copy of \(F\).

A sheaf is flabby if sections extend from any open subset to a larger one. Each \(G(F)\) is flabby, because its sections are arbitrary collections of stalk elements. Any injective is a direct summand of its own \(G(F)\), hence flabby.

We need two elementary consequences. In an exact sequence \(0\to F\to G\to H\to0\) with \(F\) flabby, a section of \(H\) lifts globally. Choose local lifts and well-order their domains. At a successor step, the old lift and the new one differ on the overlap by a section of \(F\). Extend that difference to the new domain, subtract it, and glue. At a limit step the compatible lifts glue by the sheaf axiom. If \(G\) is also flabby, this lifting assertion shows that \(H\) is flabby: lift on the smaller open, extend in \(G\), then project. Applied to successive cokernels in an injective resolution of a flabby sheaf, these assertions show that its higher derived sections vanish.

More generally a bounded-below flabby resolution computes derived sections. To see this without assuming a comparison theorem, map it into an injective resolution by the extension induction above. The cone is an exact bounded-below complex of flabby sheaves. Its cycles are flabby by induction from the first nonzero term, using the quotient assertion. Its sections are therefore exact by the lifting assertion. The comparison map on sections is a quasi-isomorphism.

For an open inclusion \(a:W\hookrightarrow X\), extension by zero \(a_!F\) consists of sections on \(O\cap W\) whose support is closed in \(O\), for each open \(O\subset X\). It has stalk \(F_x\) in \(W\) and zero outside. This proves its exactness. Extending a local morphism by zero proves \(a_!\dashv a^*\). Thus restriction to an open preserves injectives, since maps from a subobject extend after applying the exact functor \(a_!\).

A.2. A flabby cochain resolution on Euclidean open sets

For \(O\subset\mathbb R^d\) open, let \(C_q(O)\) be the vector space freely generated by continuous singular simplices \(\Delta^q\to O\), with the alternating face boundary. Put \(C^q(O;V)=\operatorname{Hom}_\Lambda(C_q(O),V)\), and let \(\mathscr C^q(V)\) be its sheafification. The cancellation of each codimension-two face twice with opposite signs gives \(\partial^2=0\), so the dual differentials give a complex of sheaves.

We first prove that every section of \(\mathscr C^q(V)\) on \(O\) is represented by one cochain on \(O\). A section has local representatives \(c_i\) on a cover \(W_i\). That cover admits a locally finite family of closed sets \(K_j\) covering \(O\), with \(K_j\subset W_{i(j)}\). Here is the needed construction. Choose compact sets

with \(E_0=E_{-1}=\varnothing\); if the complement is empty, omit the distance condition. They exhaust \(O\) and \(E_m\subset\operatorname{int}E_{m+1}\). Cover the compact layer \(E_m\setminus\operatorname{int}E_{m-1}\) by finitely many open balls whose closures lie in members of the given cover and in \(\operatorname{int}E_{m+1}\setminus E_{m-2}\). Their closures are the \(K_j\). They cover \(O\). A neighbourhood lying inside some \(\operatorname{int}E_M\) meets only finitely many layers, hence only finitely many \(K_j\).

For a singular simplex \(\sigma\), set \(c(\sigma)=c_{i(j)}(\sigma)\) using the first index \(j\) for which its image meets \(K_j\) and lies in \(W_{i(j)}\); set it to zero if there is no such index. This defines a cochain. Fix \(x\). Local finiteness and closedness allow a neighbourhood of \(x\) to avoid every \(K_j\) in that neighbourhood which does not contain \(x\). Only finitely many remaining indices matter. Shrink again inside their \(W_{i(j)}\), and until their cochains agree there: they represent the same germ at \(x\). Every simplex in this final neighbourhood has an eligible index because the \(K_j\) cover, and every eligible representative gives the same value. Thus \(c\) represents the prescribed section near \(x\), proving surjectivity.

A cochain on a smaller open extends to a larger open by assigning zero to simplices not contained in the smaller one. Surjectivity just proved therefore makes every \(\mathscr C^q(V)\) flabby. This argument uses the Euclidean hypotheses; sheafification of an arbitrary flabby presheaf need not be flabby.

The augmentation

is exact. To check a stalk, take the filtered system of convex balls about the point. On a convex set, coning a singular simplex to a fixed point \(b\) gives an operator \(p_b:C_q\to C_{q+1}\) on the augmented chain complex: in barycentric coordinates its value is

with value \(b\) at \(t_0=1\); in degree \(-1\), send \(1\) to the point \(b\). The face formula gives \(\partial p_b+p_b\partial=1\). Dualizing proves that cochains on the ball have cohomology \(V\) in degree zero and zero in positive degree. Taking the exact filtered colimit proves (A.3) on stalks. Its degree-zero kernel consists of functions constant on small path-connected balls, precisely \(V_O\).

A.3. Constant sheaves on convex regions are acyclic

We must still compute the sections of (A.3); the stalk calculation alone does not compute them. For an open cover \(\mathcal U\) of \(O\), write \(C_q^{\mathcal U}(O)\) for chains generated by simplices contained in one member. Surjectivity in A.2 and the definition of a zero germ give

where refinements induce restriction of cochains. Indeed a cochain represents zero exactly when it vanishes on every simplex in some open cover. Restriction onto the small-chain subspace is surjective by extension on the remaining basis simplices, so this describes both the quotient and the colimit.

Suppose now \(O\) is nonempty and convex. We prove that each augmented small-chain complex is exact. The subdivision operators needed for this have an explicit construction. For an affine simplex \(\lambda\), let \(b_\lambda\) be its barycenter and let \(b_\lambda*c\) mean prepend this vertex to each simplex of the affine chain \(c\). On augmented affine chains,

Define subdivision and a homotopy inductively by

The cone identity first gives \(\partial S=S\partial\). For the second formula it gives \(\partial P\lambda=\lambda-P\partial\lambda-b_\lambda*(S\partial\lambda)\), by induction on dimension. Hence \(\partial P+P\partial=1-S\). Apply these affine formulas on the standard simplex and compose with a singular simplex to define the same operators on singular chains. They preserve its image. Iteration gives

The simplices occurring in a subdivision have vertices which are barycenters of nested faces. If those faces have \(a\le b\) vertices, their barycenters differ by \((b-a)/b\) times a difference of two points of the original simplex. Thus their distance is at most \(q/(q+1)\) times its diameter in dimension \(q>0\). The diameter of a convex hull is at most the largest distance between its vertices, by expanding the difference of two convex combinations. Repeated subdivision therefore makes the mesh tend to zero. For a continuous simplex, pull back the open cover to its compact standard simplex. A finite subcover by sufficiently small balls gives a positive number such that any subset of smaller diameter lies in one cover member: choose balls of radius \(2\epsilon_x\) inside cover members, cover by finitely many balls of radius \(\epsilon_x\), and take the minimum \(\epsilon_x\). Consequently every finite singular chain becomes \(\mathcal U\)-small after sufficiently many subdivisions.

Let \(z\) be an augmented small cycle of degree at least zero; in degree zero this means its coefficients sum to zero. Its cone \(p_bz\) is a finite chain with boundary \(z\), though it need not be small. Choose \(N\) so that \(S^Np_bz\) is small. Since \(P_N\) preserves each simplex's image, \(P_Nz\) is already small. Equations (A.5)–(A.6) give

The augmentation onto \(\Lambda\) is surjective using any point of \(O\). This proves exactness in every degree. Over a field, choose a complement to the boundary subspace in each chain group; the differential identifies that complement with the next boundary subspace. This splits the augmented exact complex into two-term identity complexes. Its dual with values in any \(V\) is exact as well.

Thus each cochain complex in (A.4) has cohomology \(V\) in degree zero and zero otherwise, and refinement preserves that degree-zero identification. Exactness of filtered colimits and A.1 now prove

The same holds on a space homeomorphic to such an \(O\): transporting sheaves across a homeomorphism is an exact equivalence preserving sections and injectives. A disjoint union of two such regions has degree-zero sections \(V\oplus V\) and no higher cohomology, by the same statement on its two components.

A.4. The two angular regions and the differential

Let \(D_R^*=\{z:0<|z|<R\}\), allowing \(R=\infty\).

Transport and monodromy. Fix a base point in \(D_R^*\). A finite-rank local system is determined, including its morphisms, by its fibre \(V\) and an invertible operator \(T\) for one positive turn. We give the underlying topological and sheaf constructions.

Along a path, choose finitely many consecutive intervals mapped into trivializing opens and compose their fibre identifications. Refining intervals changes nothing. Two choices admit a common refinement; transition matrices are locally constant, so their values agree along each connected interval and cancel at its endpoints. For a homotopy, a sufficiently fine rectangular grid maps each rectangle into a trivializing open: compactness gives a Lebesgue number, by the finite-ball argument in A.3. Transport around each rectangle is the identity. Cancelling shared edges shows that endpoint-fixed homotopic paths have the same transport. Concatenation composes transport, and reversing a path inverts it. On a convex region, straight-line homotopies therefore trivialize a local system from any chosen fibre.

The covering map \((s,\theta)\mapsto e^{s+i\theta}\) has convex domain \(( -\infty,\log R)\times\mathbb R\), with \(\log R=\infty\) when needed. Local angle branches lift a path uniquely after its initial angle is fixed: subdivide into branch domains and adjust successive branches by integral multiples of \(2\pi\). The difference of two lifts is a continuous integer multiple of \(2\pi\), hence constant. The same construction works on a fine grid of a homotopy square, extending rectangle by rectangle; agreement at a corner forces agreement on each shared edge. A based loop's lifted endpoint thus differs from its starting angle by \(2\pi n\), with \(n\) invariant under based homotopy. Conversely two lifts with the same endpoints are joined by straight-line interpolation in the convex covering domain. Projecting gives a based homotopy. Concatenation adds \(n\), and the loop making \(n\) turns realizes every integer. This proves the loop classification used in Proposition 1.1.

To construct a local system from \((V,T)\), give \(V\) the discrete topology and impose on the product of the covering domain with \(V\) the identifications

Over an angular branch this quotient is the product with \(V\): every class has a unique representative in that branch. On overlaps, the coordinates differ by locally constant powers of \(T\); the powers add on triple overlaps. Continuous local sections therefore form a sheaf of vector spaces, locally the constant sheaf \(V\). Lifting one positive turn with constant coordinate \(v\) ends at the class of \((s,\theta+2\pi,v)\), which is the class of \((s,\theta,Tv)\). Its monodromy is exactly \(T\).

Conversely, pull any local system to the convex covering domain. Transport gives its constant trivialization. If \(\Phi_{s,\theta}:V\to L_{e^{s+i\theta}}\) denotes transport from the fixed base lift, the loop calculation and concatenation give \(\Phi_{s,\theta+2\pi n}=\Phi_{s,\theta}T^n\). Hence this trivialization factors through (A.11), providing an isomorphism with the original local system on every branch and thus globally. A sheaf morphism commutes with transport, so its base-fibre map satisfies \(bT=T'b\). Conversely that equation makes the constant map \(b\) respect (A.11), and it descends to a unique sheaf morphism. Fibre evaluation and the quotient construction are therefore inverse on objects and morphisms. This construction also identifies restrictions to smaller punctured discs using transport along a fixed radial ray.

For the covering calculation and its lifting proof, see Hatcher, Chapter 1, Theorem 1.7. The local-system quotient above includes the required transport compatibility.

Cover it by \(A\), with angle \(-\pi<\theta<\pi\), and \(B\), with angle \(0<\theta<2\pi\). Their intersection has upper and lower components. The coordinates \((\log|z|,\theta)\) identify each region and intersection component with an open convex rectangle. Transport trivializes \(L\) there. Choose the two trivializations to agree on the upper component. In the lower component the trivialization coming from \(B\) differs from that from \(A\) by \(T\): its continuation has made one extra positive turn.

For clarity, this cover computes derived sheaf sections. Resolve \(L\) by injectives \(I^\bullet\) on \(D_R^*\). In every degree the sequence

is exact: the kernel assertion is sheaf gluing, and a section on the intersection extends to \(B\) because \(I^q\) is flabby. The total complex of its last two columns is therefore quasi-isomorphic to \(\Gamma(D_R^*,I^\bullet)\). Restrictions remain injective by A.1; (A.8) shows each of these columns has only degree-zero cohomology, equal to the sections of \(L\). Replacing those columns by their degree-zero section complexes is a quasi-isomorphism on totals. To verify this last assertion, filter the cone of the comparison by its two columns; the two successive quotients are shifted acyclic column cones, so the total cone is acyclic.

Consequently the required derived sections are represented by

Use domain coordinates \((z,y)=(y-x,y)\) and target coordinates \((s,t-s)\). The differential becomes

Discarding the contractible identity summand gives \([V\xrightarrow{T-1}V]\) in degrees \(0,1\). All maps commute with intertwiners \(bT=T'b\), so this is a functorial derived model. Shrinking \(R\) restricts the same angular cover and the same locally constant sections; identifying base fibres along one radial ray makes the resulting two-term maps identities. This proves the compatibility needed for the stalk colimit in Proposition 1.1, without an unproved homotopy-invariance assertion for sheaf cohomology.

The familiar cellular description agrees with this model. The augmented chains of the lifted one-vertex, one-edge circle are \(0\to\Lambda[t,t^{-1}]\xrightarrow{t-1}\Lambda[t,t^{-1}]\xrightarrow{f\mapsto f(1)}\Lambda\to0\). They are exact: multiplication by \(t-1\) is injective, and division after clearing negative powers identifies the augmentation kernel with its image. Applying equivariant \(\operatorname{Hom}(-,V)\) gives exactly the two-term complex already obtained from (A.9)–(A.10). The derived-sheaf justification is the preceding argument.

For cochain sheaves and their topological hypotheses, see Sella; for subdivision, see Hatcher; for flabby sheaves, see the Stacks Project references below.

Appendix B. Classical localization and the local curve calculations

This appendix works with classical sheaves of vector spaces over a field. It constructs the bounded-below derived operations needed for the boundary, stalk and costalk calculations of this lesson. It also checks the local branch model and the explicit normalization of the displayed nodal cubic. Appendices F–K subsequently prove the general classical operations, duality and algebraic constructibility. Appendix L proves the general node and normalization-to-branch comparison. The étale/adic foundations remain separate.

B.1. Complexes, resolutions and triangle signs

For complexes, use

Thus degree-zero cocycles are chain maps and boundaries are chain homotopies. Appendix A.1 gives the exact injective functor \(G(F)=\prod_x(e_x)_*F_x\) and its natural monomorphism \(F\to G(F)\). Exactness of \(G\) can be tested even on sections: its maps are products of exact stalk sequences of vector spaces. Let \(C(F)\) be the cokernel of this monomorphism. The diagram of the two exact rows \(0\to F\to G(F)\to C(F)\to0\), applied to a short exact sequence of \(F\)'s, shows that \(C\) is exact: injectivity at the left follows by subtracting a lift from \(F\), surjectivity at the right by lifting in \(G(F)\), and the same subtraction proves equality of image and kernel in the middle. This is a stalkwise argument.

The functors \(G C^q\), for \(q\ge0\), therefore give an additive functorial injective resolution of every sheaf. For a complex \(E\) zero below degree \(a\), apply these resolutions to every \(E^p\). Totalize with the differential \(d_E+(-1)^p d_{\mathrm{res}}\) on the component \(G C^q(E^p)\). In total degree \(n\), only \(a\le p\le n\) occur, so every term is a finite sum of injectives. Denote this complex by \(I(E)\).

The map \(E\to I(E)\) is a quasi-isomorphism. In its augmented double complex, the resolution direction is exact. For a total cocycle, begin with its smallest original degree \(p\). Exactness in the resolution direction allows that component to be removed by a boundary. Proceed to the next \(p\). There are finitely many components on the diagonal; at the final augmented position injectivity of \(E^p\to G(E^p)\) makes the remaining component zero. This proves exactness of the augmented total complex. The construction sends a homogeneous map to its componentwise images under \(G C^q\); the vertical terms cancel in (B.1), so it respects differentials of Hom complexes and compositions.

Hom into a bounded-below injective complex sends acyclic complexes to acyclic complexes. Here is the required homotopy construction. For an acyclic \(N\) and a chain map \(f:N\to I\), start below the first nonzero degree of \(I\). Suppose the homotopy equations hold below \(n\). The map \(f^n-d_Ih^n\) vanishes on \(\ker d_N^n=\operatorname{im}d_N^{n-1}\), using the previous equation and \(d_If=fd_N\). It therefore defines a map on \(\operatorname{im}d_N^n\) by \(h^{n+1}(d_Nx)=f^n(x)-d_Ih^n(x)\). Injectivity of \(I^n\) extends it to \(N^{n+1}\), giving the next equation. Induction yields \(f=d_Ih+hd_N\). Apply this in every shift to see that the whole Hom complex is acyclic. Applying it to the acyclic cone of a quasi-isomorphism proves that Hom into \(I\) preserves that quasi-isomorphism. In particular an acyclic bounded-below injective complex is contractible, by applying that assertion to its identity. Hence a quasi-isomorphism between such complexes is a homotopy equivalence. Explicitly, in a contraction of its cone the lower-left block is a chain map back; the two diagonal equations say that both composites are homotopic to the identity. The homotopy category of these injective complexes realizes the localization of bounded-below complexes at quasi-isomorphisms: replace by \(I(E)\), which inverts those maps, and use the natural quasi-isomorphism \(E\to I(E)\). On injective complexes that natural map is already a homotopy equivalence. This proves the claimed universal localization property and gives the usual \(D^+(X)\).

For clarity about the triangle input, our cone is \(\operatorname{Cone}(f)^n=F^n\oplus E^{n+1}\), with \(d(y,x)=(d_Fy+fx,-d_Ex)\). Cone triangles use the inclusion of \(F\) and projection to \(E[1]\). Their triangle identities can be checked inside the homotopy category. The cone of an identity contracts by \((y,x)\mapsto(0,y)\). The cone of the inclusion \(F\to\operatorname{Cone}(f)\) projects to \(E[1]\) with a contractible identity summand; a section sends \(x\) to \((0,x,-fx)\), giving the rotated last map \(-f[1]\). A square with \(bf-f'a=dh\) induces the cone map \((y,x)\mapsto(by+hx,ax)\); this verifies the triangle-map axiom, including squares commuting only up to homotopy.

For the octahedron of \(E\xrightarrow fF\xrightarrow gH\), take

The cone of the first arrow has coordinates \((z,x,y,x')\). Its projection \((z,x,y,x')\mapsto(z,y+fx)\) onto \(\operatorname{Cone}(g)\) is a chain map with section \((z,y)\mapsto(z,0,y,0)\). In the coordinates \((z,y+fx,x,x')\), the remaining summand has differential \((x,x')\mapsto(-dx+x',dx')\) and contracts by \((x,x')\mapsto(0,x)\). The resulting last arrow is \((z,y)\mapsto(y,0)\) into \(\operatorname{Cone}(f)[1]\). Write the first two arrows of (B.2) as \(\alpha,\beta\), and write cone inclusions and projections as \(\iota,p\). The required squares are the identities \(\alpha\iota_f=\iota_{gf}g\), \(p_{gf}\alpha=p_f\), \(\beta\iota_{gf}=\iota_g\), and \(p_g\beta=f[1]p_{gf}\). The last arrow is \(\iota_f[1]p_g\). The composite \(\beta\alpha\) contracts by \((y,x)\mapsto(0,y)\): its homotopy differential is \((gy,fx)\). These are the octahedron identities and its cone triangle, proving the remaining triangle axiom. Shifts and cones preserve bounded-below injective complexes, so these verifications apply to \(D^+(X)\).

The ordinary t-structure also follows from complexes. The good lower cut uses a kernel in its final degree, and the good upper cut uses a cokernel in its first degree. Their cohomology is respectively the original cohomology below and above the cut, with zero cohomology on the other side; the cut triangle follows from the quotient complex and its direct kernel/image calculation. If one object has cohomology in degrees \(\le0\), represent it by its lower cut with zero terms above zero. Resolve the upper cut of an object in degrees \(\ge1\) by the construction above, with zero terms below one. There is no degree-zero chain map between these representatives. This proves orthogonality and all ordinary t-structure axioms used below.

For later use, these models also give long exact cohomology and section sequences. A short exact sequence of sheaves becomes a termwise short exact sequence under \(I\), since every \(G C^q\) is exact. Each resulting sequence of injectives splits on terms because its left term is injective; applying sections therefore leaves an exact sequence of complexes. The connecting map sends a quotient cocycle to the differential of any lift, regarded in the subcomplex. Changing the lift changes this by a boundary, and lifting or subtracting a coboundary proves exactness at each of the three kinds of term. The same argument on the termwise split sequence of any cone proves its long cohomology sequence. These arguments justify the cohomology sequences used in the body of the lesson.

B.2. The actual open-closed recollement

Let \(i:F\hookrightarrow X\) be closed and \(j:U\hookrightarrow X\) its open complement. Ordinary inverse image \(i^*\), open restriction \(j^*\), closed pushforward \(i_*\) and open extension by zero \(j_!\) are exact: their stalks are the indicated original stalks or zero. Their adjunctions follow by restricting a morphism and gluing its forced zero values outside the relevant support. For closed pushforward, opens of \(F\) are intersections with opens of \(X\), which also proves \(i^*i_*=1\). For open direct image, sections on \(O\) are sections on \(O\cap U\), giving \(j^*\dashv j_*\) and \(j^*j_*=1\).

There is a stalkwise short exact sequence of sheaves

It gives the first triangle in (B.5) below. More explicitly, for a monomorphism of complexes \(A\to B\), the cone projects quasi-isomorphically onto the quotient complex: the kernel is an identity cone on \(A\). Thus this use of a short exact sequence really gives a cone triangle, even if the sequence does not split on terms.

The exact left adjoints above show that \(j_*\) and \(i_*\) preserve injectives, and exact \(j_!\) shows that \(j^*\) preserves injectives. Consequently the chain adjunctions, applied to injective targets, give the derived adjunctions, including \(j^*\dashv Rj_*\). The identity \(j^*Rj_*=1\) follows on an injective resolution by restriction.

There is also an ordinary support right adjoint \(i^!_{\mathrm{sh}}\) to \(i_*\). Define \(i_*i^!_{\mathrm{sh}}E\) to be the kernel of \(E\to j_*j^*E\). A sheaf zero on \(U\) is identified with its closed pushforward by (B.3), so this defines a sheaf on \(F\). Any map from \(i_*A\) lands in that kernel, since \(\operatorname{Hom}(i_*A,j_*j^*E)=\operatorname{Hom}(j^*i_*A,j^*E)=0\). This proves the right adjunction. Since \(i_*\) is exact, \(i^!_{\mathrm{sh}}\) sends injectives to injectives.

An injective \(I\) is flabby by A.1. Its sections on \(O\cap U\) extend to \(O\), so

is exact, even on sections. Resolve a complex by injectives and apply this sequence degreewise. Since \(j^*I\) is an injective resolution on \(U\), (B.4) gives the second triangle (B.6) below with \(i^!=Ri^!_{\mathrm{sh}}\), and its chain adjunction proves \(i_*\dashv i^!\). Restriction of an extension gives \(j^*j_!=j^*Rj_*=1\) and \(i^*i_*=1\). Resolving on \(F\) and pushing forward gives \(i^!i_*=1\), while disjoint support gives \(j^*i_*=i^*j_!=0\). On an injective open model \(J\), the map \(j_*J\to j_*j^*j_*J\) is the identity, so its support kernel is zero and \(i^!Rj_*=0\). These unit and counit identities prove full faithfulness of \(j_!,Rj_*\) and \(i_*\): the corresponding adjunction identifies maps between two extensions with the maps between their restrictions. All these operations preserve shifts and cone triangles on the models just constructed. This establishes recollement in \(D^+\), allowing complexes unbounded above.

Explicitly, the two triangles are

B.3. Point support, local excision and the smooth-disc costalk

Write \(i_p:\{p\}\hookrightarrow X\) for a closed point and \(j:X\setminus\{p\}\hookrightarrow X\). Applying exact \(i_p^*\) to the support triangle gives the natural model

Here the fiber of \(f:E\to F\) has terms \(E^n\oplus F^{n-1}\) and differential \(d(x,y)=(d_Ex,fx-d_Fy)\). Its projection to \(E\) and connecting map to \(F\) give the support triangle with the corresponding cone sign convention. Formula (B.7) determines the object and its restriction map, not just the dimensions of its groups.

It also computes cohomology supported at \(p\) on every open neighbourhood \(W\). For an injective sheaf \(I\), put \(\Gamma_p(W,I)=\ker(\Gamma(W,I)\to\Gamma(W\setminus\{p\},I))\). Flabbiness makes the latter restriction surjective. A section supported at \(p\) extends uniquely by zero from \(W\) to any larger neighbourhood, by gluing on \(W\) and the complement of \(p\). Thus \(\Gamma_p(W,I)=i_p^!{}_{\mathrm{sh}}I\), independently of \(W\). Restriction to \(W\) preserves injectives by A.1, so the degreewise exact section sequence on an injective resolution proves

This is the sheaf-theoretic relative-section calculation and its excision statement. It has been proved from resolutions; no assertion of homotopy invariance of arbitrary sheaf cohomology is being inserted.

For a smooth complex disc and its centre, A.3 gives \(R\Gamma(D,\Lambda)=\Lambda[0]\), and A.4 gives \(R\Gamma(D^*,\Lambda)=[\Lambda\xrightarrow0\Lambda]\) in degrees zero and one. The restriction map sends a constant to the same constant in degree zero. Its fiber is

Cancel the identity pair: the costalk is \(\Lambda[-2]\). Formula (B.8) also proves that this is the costalk in any larger curve neighbourhood with the same local sheaf and disc chart. For \(j_!L\), the point stalk is zero and the boundary term is Proposition 1.1, so (B.7) gives its shift by \([-1]\). For \(Rj_*L\), the boundary restriction map is an identity, so the fiber is zero. These are all the classical assertions of Proposition 1.2.

B.4. Finitely many smooth branches

Let \(X_r\) be \(r\) open complex discs with their centres identified to a point \(p\), with the quotient topology, and let \(k_a:D_a\hookrightarrow X_r\) be the branch inclusions. Each branch is closed. Its complement is the union of the other punctured discs, which is open; conversely the quotient topology restricts to the usual disc topology. The map from the disjoint union of the discs separates the branches. Its pushforward of the constant sheaf is \(N=\bigoplus_{a=1}^r(k_a)_*\Lambda_{D_a}\), since sections on a disjoint union are products of sections on the components and here there are finitely many components.

The constant-sheaf restriction to all branches gives an exact sequence

The formula describes the last map on an open containing \(p\); elsewhere it is zero. It is a sheaf morphism, and exactness follows on stalks: away from \(p\) the first map is the identity on its branch, and at \(p\) it is the diagonal \(\Lambda\to\Lambda^r\). Each small neighbourhood of the centre is connected, so the constant-sheaf stalk is indeed \(\Lambda\). The difference map at that stalk is surjective with precisely the diagonal kernel.

Closed pushforward is exact and preserves injectives by B.2. Thus derived sections of each branch pushforward are derived sections of its disc; by A.3 they are \(\Lambda\) in degree zero. The point term is also acyclic. Applying derived sections to (B.10) therefore proves \(R\Gamma(X_r,\Lambda)=\Lambda[0]\), with its specified constant-section identification: the complex is the fiber of the surjective difference map \(\Lambda^r\to\Lambda^{r-1}\). The same argument works after decreasing the radius of each disc. It justifies the neighbourhood acyclicity used in the relative-section interpretation.

Alternatively compute the costalk directly from (B.7). The puncture is a disjoint union of punctured discs, so its boundary complex is \([\Lambda^r\xrightarrow0\Lambda^r]\). This identification is compatible with decreasing radii by A.4 and with the exact stalk colimit by A.1. For \(\Lambda_{X_r}\), the attaching map is diagonal in degree zero. Therefore

Its cohomology is \(\Lambda^{r-1}\) in degree one and \(\Lambda^r\) in degree two, with all other groups zero. For \(N\), the point stalk is \(\Lambda^r\) and the same attaching map is the identity on \(\Lambda^r\). Its fiber cancels that identity pair and leaves \(\Lambda^r[-2]\). This proves the normalization costalk in the branch model using exact sheaf operations alone. The case \(r=1\) is (B.9); \(r=2\) is Proposition 1.3.

For an ordinary complex node we use its analytic local model \(xy=0\); its two coordinate discs give exactly \(X_2\). For an ordinary multiple point with smooth branches and distinct tangent lines, the same topological model has \(r\) branches. Here is the local topological verification. Choose analytic disc coordinates on each smooth branch. Each branch parametrization is an embedding, and for two distinct tangent lines its two first-order tangent vectors are linearly independent over \(\mathbb C\). Thus the norm of the difference of their linear parts is bounded below by \(c(|s|+|t|)\) for some \(c>0\): restrict that injective linear map to the compact unit sphere. The higher-order remainders are bounded by a quantity smaller than \(c(|s|+|t|)/2\) after shrinking the parameters. The branches therefore meet only at the common centre. A finite union of smaller closed parameter discs maps continuously and bijectively to its image, and the inverse is continuous because the domain is compact and the ambient complex analytic space is Hausdorff: images of closed subsets are compact, hence closed. Restricting to the interiors gives the asserted local quotient topology. The disc charts used here are part of the smooth-branch hypothesis. L.3–L.5 prove that the algebraic normalization separates exactly these smooth branches and is a local analytic isomorphism on each; L.2–L.4 derive the two-branch model from the completed-local node criterion. The explicit cubic below also verifies its branch description directly.

B.5. The displayed cubic and its actual normalization

For \(X:\ y^2z=x^2(x+z)\subset\mathbb P^2_\mathbb C\), define

The three coordinates never vanish together, and direct substitution verifies the cubic equation. On \(z\ne0\), write \(t=u/v\). The map is \(x=t^2-1\), \(y=t(t^2-1)\). It identifies the affine curve ring with the subring

To check that there are no extra relations, divide by the monic polynomial \(y^2-x^2(x+1)\). A possible kernel element has the form \(P(x)+yQ(x)\). After substitution its first summand is even in \(t\), and its second is odd. Both must vanish; hence \(P=Q=0\). The function fields agree, because \(t=y/x\) where \(x\ne0\). The element \(t\) satisfies \(t^2=x+1\), so \(\mathbb C[t]\) is a finite \(A\)-module generated by \(1,t\).

The polynomial ring \(\mathbb C[t]\) is integrally closed in its fraction field. Indeed, write an integral rational function as \(p/q\) with coprime polynomials. Multiplying a monic integral equation by \(q^n\) shows that \(q\) divides \(p^n\). Polynomial division and the Bézout identity then force \(q\) to be a nonzero constant. Any element integral over \(A\) is also integral over \(\mathbb C[t]\), using the same monic equation; hence it is in \(\mathbb C[t]\). Conversely reduce a polynomial in \(t\) to \(a+bt\) with \(a,b\in\mathbb C[x]\subset A\). This element satisfies the monic equation \((Z-a)^2-b^2(x+1)=0\), so every element of \(\mathbb C[t]\) is integral over \(A\). Thus this is the actual normalization on the affine chart.

At infinity use the chart \(y\ne0\) and coordinates \(\xi=x/y\), \(\zeta=z/y\). Its equation is \(\zeta(1-\xi^2)=\xi^3\). The factor \(1-\xi^2\) is a unit: in this ring \((1-\xi^2)(1+\xi^2+\xi\zeta)=1\). The chart ring is therefore \(\mathbb C[\xi,(1-\xi^2)^{-1}]\), which is integrally closed by the same coprime-denominator argument with the inverted factors allowed. On its preimage, \(s=v/u\) is a coordinate, \(s^2\ne1\), and (B.12) gives \(\xi=s\), \(\zeta=s^3/(1-s^2)\). Thus \(\nu\) is an isomorphism on this chart. These two charts cover \(X\), proving that (B.12) is finite and is its normalization. In particular the normalization is the smooth rational curve \(\mathbb P^1\).

The two preimages of \(p=[0:0:1]\) are \(t=1,-1\). For \(|x|<1\), the convergent binomial series \(h(x)=(1+x)^{1/2}\), with \(h(0)=1\), has \(h(x)^2=1+x\): convergence follows from the ratio test and the identity by multiplying its power series. The two branches of the cubic are exactly \(y=xh(x)\) and \(y=-xh(x)\). They are graphs over the \(x\)-disc, have distinct tangent lines \(y=x\) and \(y=-x\), and meet only at zero after shrinking because \(h\) is nonzero there. The map in (B.12) restricts to their inverse parametrizations \(t=\pm h(x)\). This verifies the branch model and the two-dimensional normalization stalk for the concrete cubic used in Section 1 (the node calculation).

Finally, this cubic has no other singularities. On \(z=1\), the derivatives of \(y^2-x^3-x^2\) are \(2y\) and \(-3x^2-2x\); their simultaneous zeros on the curve give only \((0,0)\). At its sole point at infinity, \([0:1:0]\), the derivative with respect to \(z\) of the homogeneous equation is \(y^2-x^2\ne0\). Consequently the single-point sequence (1.2) is global for this cubic. On a curve with several nodes it instead holds locally at the chosen node; globally the quotient has a summand at each node, obtained from the same stalkwise difference maps.

Appendix C. Sheaf cohomology of the cone and local orientation groups

This appendix supplies the comparison needed in Section 1 (the cone calculation). Throughout it, coefficients are a field. The spaces considered below are locally compact, second countable and metrizable, and have a basis of contractible open neighbourhoods. The cone neighbourhood, its puncture, its link, and their open subsets satisfy these hypotheses by the explicit geometry in C.3. A deformation retraction is used only after the comparison with singular cochains has been proved.

C.1. The prism and small chains

For a homotopy \(H:Y\times[0,1]\to Z\), write \(v_i=(e_i,0)\) and \(w_i=(e_i,1)\) in the prism over a standard \(q\)-simplex. On a singular simplex \(\sigma\), define

The notation for a bracket means the affine map from the standard simplex with that ordered list of vertices. Expanding the face boundary gives

Here is the cancellation. Deleting a vertex other than the duplicated index \(i\) gives a simplex in the prism over that face, with the negative of its coefficient in \(Q_H\partial\). Deleting either occurrence of the duplicated index gives an interior face shared with the adjacent prism simplex; its two coefficients are opposite. The two unpaired terms are the all-upper face with coefficient \(+1\) and the all-lower face with coefficient \(-1\). This proves (C.2) in every degree, including degree zero. It also proves the formula for pairs whenever the homotopy preserves the subspace. Dualizing gives the corresponding cochain homotopy. In particular, a contraction identifies cohomology with that of a point. The point's augmented chain complex is exact directly: it has one basis simplex in each degree, with boundary alternately zero and identity, and augmentation the identity in degree zero.

For any open cover \(\mathcal U\) of any space \(Y\), the inclusion of small chains

is a quasi-isomorphism. This follows from the actual operators (A.5)–(A.6), without convexity of \(Y\). A finite cycle \(z\) becomes small after enough subdivisions, and \(z-S^Nz=\partial P_Nz\), proving surjectivity on homology. If a small cycle bounds an arbitrary chain, \(z=\partial b\), choose \(N\) making \(S^Nb\) small. Then \(z=\partial(S^Nb+P_Nz)\), and both summands in this filling are small: the second preserves the images of the already small simplices of \(z\). This proves injectivity. The same proof works for augmented chains.

Dualizing (C.3) over a field remains a quasi-isomorphism. Indeed its cone is an exact complex of vector spaces. Choose complements to the cycle subspaces degree by degree; the differentials identify the complementary pieces with the next cycle subspaces. This gives a contraction of the cone. Applying \(\operatorname{Hom}_\Lambda(-,V)\) to that contraction proves exactness of the dual cone for every coefficient vector space \(V\). Thus this step does not assume finite-dimensional chain groups.

C.2. The actual sheaf–singular comparison

Let \(Y\) have the hypotheses stated above. Every open \(O\subset Y\) admits a compact exhaustion \(E_m\subset\operatorname{int}E_{m+1}\). To construct one, choose a countable cover by relatively compact open sets \(B_m\). Such a cover exists because local compactness and the metric give a precompact neighbourhood inside any prescribed neighbourhood, and a second countable space reduces any open cover to a countable one: for each basic open set contained in a cover member, choose one such member. Inductively take a finite union of relatively compact open sets covering \(E_m\cup\overline B_{m+1}\), and let \(E_{m+1}\) be its compact closure. Beginning with \(E_0=\varnothing\) gives the asserted exhaustion of \(O\).

Put \(E_{-1}=E_0=\varnothing\). For any prescribed open cover of \(O\), cover each compact layer \(E_m\setminus\operatorname{int}E_{m-1}\) by finitely many neighbourhoods whose closures lie in a cover member and in \(\operatorname{int}E_{m+1}\setminus E_{m-2}\). The metric permits this shrinking. Those closures give a locally finite closed refinement: a neighbourhood inside \(\operatorname{int}E_M\) meets no layer chosen after \(M+2\). This proves the refinement property on every open subset, not just on \(Y\).

Sheafify the singular cochain presheaves as in A.2, writing \(\mathscr C_Y^q(V)\). The representative argument there now applies with this refinement: assign a simplex the first eligible local representative, and near a point exclude all the finitely many relevant closed sets not containing it. The remaining representatives agree after shrinking because they have the same germ. Every sheaf section is therefore represented by one cochain. Extending that cochain by zero on other basis simplices proves that \(\mathscr C_Y^q(V)\) is flabby.

On a contractible neighbourhood, (C.1)–(C.2) and the point complex show that augmented cochains are exact. Exact filtered colimits therefore prove stalkwise exactness of

Its degree-zero kernel is the locally constant functions: a zero coboundary makes values equal along every path in a small contractible neighbourhood. Thus (C.4) is a flabby resolution of the constant sheaf, and A.1 shows that its sections compute derived sheaf sections.

The same representative and zero-germ arguments as in (A.4) identify those sections on every \(O\) with

By (C.3), the restriction map from all singular cochains to each complex on the right is a quasi-isomorphism. Refinements commute with those maps. Exactness of filtered colimits, proved in A.1, now gives the natural quasi-isomorphism

It commutes with restriction to smaller open sets. It also respects continuous pullback wherever both spaces have these hypotheses: composing a singular simplex with the map defines pullback on cochains and on their germs, and the latter agrees with the pullback of the constant sheaf. The injective comparison in A.1 identifies that map with the derived section map. Consequently the prism proves homotopy invariance for these constant-sheaf cohomology calculations. No such assertion is made for arbitrary sheaves.

For an open \(O'\subset O\), the restriction of singular cochains is surjective on terms because the simplices in \(O'\) form a subset of the basis simplices in \(O\). Its kernel is the relative cochain complex \(\operatorname{Hom}(C_\bullet(O)/C_\bullet(O'),V)\). The termwise exact sequence identifies this kernel with the fiber of the restriction map; the identity-cone argument in B.2 proves that identification. Taking the two natural quasi-isomorphisms (C.6) identifies this relative complex with the relative derived-section fiber. In particular, when \(O'=O\setminus\{p\}\), (B.8) proves the promised relative singular-cohomology interpretation of the costalk.

C.3. Radial geometry and the vertex complex

Give the tautological line bundle \(L=\mathcal O_C(-1)\subset C\times\mathbb C^3\) its induced Hermitian norm, and write \(E=\{(x,u)\in L:\|u\|=1\}\). The maps

are continuous inverses, where \(B_R=X\cap\{\|z\|<R\}\). The line-bundle description itself is an explicit isomorphism away from zero: a nonzero cone vector spans its point of \(C\), and the tautological vector maps back to itself. In a projective coordinate chart these are the usual coordinate ratios, so they are continuous (and algebraic in the line-bundle statement).

The link of radius \(\epsilon\) is identified with \(E\) by \((x,u)\mapsto\epsilon u\). In a local holomorphic frame \(s(x)\), its coordinates are \((x,e^{i\theta})\mapsto e^{i\theta}s(x)/\|s(x)\|\). These show that it is a smooth three-manifold. Its orientation is the complex orientation of the base followed by the increasing angular coordinate. A change of frame adds a smooth angle locally; the coordinate Jacobian has positive base determinant and angular derivative \(1\), so the orientations agree. The link is compact because it is a closed subset of the compact sphere.

Collapsing \(E\times\{0\}\) in \(E\times[0,R)\) gives exactly \(B_R\) topologically. Only continuity of the inverse at the collapsed point needs checking. An open subset of the product containing \(E\times\{0\}\) contains \(E\times[0,\delta)\) for some \(\delta>0\): cover compact \(E\) by finitely many product neighbourhoods and take their minimum positive radius. These are precisely the small radial neighbourhoods of \(v\). The contraction \(z\mapsto(1-t)z\) proves contractibility of \(B_R\).

All spaces just used are metrizable and second countable as subspaces or local products of the displayed Euclidean charts. They are locally compact. Away from the vertex, those charts give contractible neighbourhood bases; at the vertex the sets \(B_\delta\) do. Thus C.2 applies. In particular \(R\Gamma(B_R,\Lambda)=\Lambda[0]\).

Let \(p_R:B_R\setminus\{v\}\to E\) be the projection in (C.7). It has section at radius \(R/2\); the homotopy \((u,r,t)\mapsto(u,(1-t)r+tR/2)\) stays in that puncture and contracts each fiber. Equations (C.2) and (C.6) prove that \(p_R^*\) is a quasi-isomorphism on cochains and computes derived sheaf sections. The projections agree under restriction to every smaller radius. Hence the exact stalk colimit on an injective resolution gives a compatible object identification

The first arrow sends a scalar to the constant zero-cochain. Indeed that is the restriction of a constant section on every \(B_R\); the comparisons respect those sections and the shrinking maps. Thus (C.8) identifies the attaching map as well as its source and target objects. The fiber's cohomology is

with the usual augmented meaning of reduced cohomology. For connected \(E\), degree one vanishes. Equations (C.8)–(C.9) prove the vertex comparison used in Proposition 1.5. They still require the separate circle-bundle and curve-cohomology calculation of Proposition 1.4 for its numerical dimensions. Finally \(i_v^!Rj_*\Lambda=0\) is the already proved injective support-kernel calculation in B.2; it needs no additional homotopy assertion.

C.4. Spheres, local orientation and coordinate changes

The same constructions determine the local orientation groups in every real dimension. Cover \(S^d\), for \(d\ge1\), by the complements of its north and south poles. Stereographic projection identifies either open with \(\mathbb R^d\); the intersection is \(S^{d-1}\times(-1,1)\), by separating the last coordinate and normalizing the other coordinates. The two-open-cover exact sequence on injectives in (A.9), together with C.2, gives the reduced Mayer–Vietoris sequence. Both opens are contractible, so its connecting map is

For \(S^0\), the two points give reduced degree-zero cohomology \(\Lambda\) and zero otherwise. Induction proves that \(\widetilde H^k(S^d;\Lambda)\) is \(\Lambda\) for \(k=d\) and zero otherwise. This also proves finiteness here without a triangulation theorem.

The puncture of \(\mathbb R^m\) retracts radially onto \(S^{m-1}\). Its ambient space is contractible, so B.8 and C.2 give

For \(m=0\) the point functor is the identity. A choice of ordered real coordinates fixes a generator in (C.11) by the connecting maps above. Reflecting one coordinate acts by \(-1\). In dimension one this is the swap of the two puncture components on \(\Lambda^2/\Lambda\). In higher dimensions, use that coordinate as the polar coordinate in the sphere cover. Reflection swaps the two opens, while its action on their intersection retracts to the identity on the equator. The cover differential is a difference of restrictions; swapping the order negates it, and hence negates the connecting generator. This proves the sign assertion in all dimensions.

Every invertible real linear map of positive determinant is joined to the identity through invertible maps. One explicit reduction is Gram–Schmidt: it writes the matrix as \(QR\), with positive diagonal in the upper triangular \(R\) and \(Q\in SO(m)\). The path from \(R\) to its diagonal obtained by decreasing the entries above it stays invertible, and its positive diagonal entries can then be moved to \(1\). Successive rotations in coordinate two-planes move the columns of \(Q\) to the standard columns, keeping the preceding columns fixed; the last column has positive sign because the determinant is one. These rotations give a path from \(Q\) to the identity. A negative-determinant map is a coordinate reflection times a positive-determinant one. The prism therefore shows that a linear coordinate change acts on (C.11) by the sign of its determinant.

For a differentiable change of coordinates fixing zero, write \(\phi(x)=Ax+r(x)\), with \(A\) invertible and \(\|r(x)\|/\|x\|\to0\). There is \(c>0\) with \(\|Ax\|\ge c\|x\|\), by compactness of the unit sphere. On a sufficiently small ball, all maps \(Ax+t r(x)\) avoid zero away from zero. They give a homotopy of pairs to the linear map. Relative comparison and (C.2) therefore prove the same determinant rule for the local cohomology generator. Its signs are locally constant on chart overlaps and multiply under composition; they define the orientation local system.

For a complex-linear coordinate change \(A\), its real determinant is \(\det_{\mathbb C}A\,\overline{\det_{\mathbb C}A}=|\det_{\mathbb C}A|^2>0\). To verify the formula, complexify the underlying real matrix: in the complex coordinates \((z,\bar z)\) it has the two diagonal blocks \(A,\bar A\); determinants are invariant under that change of basis. Holomorphic charts therefore give a consistent real orientation.

C.5. Compact support on an open ball

For a locally compact Hausdorff space and a sheaf \(F\), compactly supported sections are \(\Gamma_c(Y,F)=\varinjlim_K\Gamma_K(Y,F)\), over compact subsets \(K\). Apply this equality to an injective resolution. Filtered colimits of vector spaces are exact, and the resolution computes every derived support functor, so

For a closed compact \(K\), flabbiness gives the termwise surjective restriction to \(Y\setminus K\). Its kernel is \(\Gamma_K\), so the fiber formula (B.8) holds with \(K\) in place of the point. If \(K\subset W\subset Y\) with \(W\) open, a section supported in \(K\) extends uniquely by zero; the injective restriction argument gives excision for these supported derived sections as well.

In \(Y=\mathbb R^m\), \(m\ge1\), the centered closed balls \(K_R\) are cofinal among compact subsets. Their complements are \(S^{m-1}\times(R,\infty)\). C.2 and C.10 identify the support fiber with \(\Lambda[-m]\). Increasing \(R\) leaves the sphere projection and constant-section map unchanged, so the induced maps on the sole nonzero group are identities in these coordinates. Equation (C.12) proves

The assertion for \(m=0\) is immediate. Transport across a chart proves the same statement for an open ball. In particular its linear dual has its unique nonzero cohomology in degree \(-m\). These are the compact-support and orientation calculations needed in Proposition 2.3. Appendix F constructs the global right adjoint and its compact-support pairing and identifies these local duals with its restriction; the compact ball calculation is one of that construction's proved inputs.

For cochain-sheaf comparison, see Sella; for the prism and subdivision constructions, see Hatcher.

Appendix D. Integral Thom classes and the circle-bundle sequence

This appendix proves the classical Gysin sequence used in Section 1 (the cone calculation) and Exercise 5. It works with an arbitrary abelian coefficient group \(G\), in particular \(\mathbb Z\) or the field \(\Lambda\). The vector bundles in the theorem have a finite trivializing open cover; a bundle on the compact curve in Section 1 (the cone calculation) has such a cover. No assertion about étale Gysin maps follows from this topological argument.

D.1. Integral small chains, pairs and covers

Write \(C_\bullet(Y)\) for the free abelian singular chain complex, and put \(C_\bullet(Y,A)=C_\bullet(Y)/C_\bullet(A)\). These relative groups are free: their basis consists of the simplices not wholly contained in \(A\). Cochains with values in \(G\) are \(\operatorname{Hom}_{\mathbb Z}(C_\bullet(-),G)\), with \((da)(c)=a(\partial c)\). The prism, subdivision and homotopy formulas (C.1)–(C.2) and (A.5)–(A.6) have integer coefficients. Their face cancellations and mesh estimates therefore hold over \(\mathbb Z\) without change.

To dualize the small-chain comparison with arbitrary coefficients, we need more than the field argument in C.1. For a fixed open cover \(\mathcal U\), construct a retraction \(\rho:C_\bullet(Y)\to C_\bullet^{\mathcal U}(Y)\) and an operator \(D:C_n(Y)\to C_{n+1}(Y)\) satisfying

where \(\iota\) is inclusion. Induct over the dimension of each basis simplex \(\sigma\), starting with \(D=0\) in negative degrees. Suppose the construction is made on its faces. Set \(z=\sigma-D\partial\sigma\); the induction identity gives \(\partial z=\rho\partial\sigma\), which is small. Choose \(N\) so that \(S^Nz\) is small and define

The second term in \(\rho\sigma\) is small because \(P_N\) preserves simplex images. The identity for \(P_N\) gives both \(\partial\rho\sigma=\rho\partial\sigma\) and \(\partial D\sigma=\sigma-D\partial\sigma-\rho\sigma\). If \(\sigma\) is already small, its faces have \(D=0\) by induction, and we choose \(N=0\). Thus \(\rho\sigma=\sigma\) and \(D\sigma=0\). Every term in (D.2) remains in the image of \(\sigma\), so these operators preserve chains in every subspace. They descend to relative chains. Applying \(\operatorname{Hom}(-,G)\) to (D.1) proves the absolute and relative small-cochain comparisons for every \(G\).

The pair sequence follows directly from the degreewise split chain sequence \(0\to C_\bullet(A)\to C_\bullet(Y)\to C_\bullet(Y,A)\to0\): restriction of cochains is onto by extending their values on basis simplices. Its connecting map sends a cocycle on \(A\) to the differential of any extension on \(Y\), regarded as a relative cocycle. Changing the extension changes this by a relative coboundary. This construction proves exactness: a class killed by the next map is represented, after subtracting a differential of a lift, by a cocycle in the preceding complex. The same lifting argument gives the homology long exact sequence.

For \(Y=U\cup V\), there is also a degreewise split exact sequence

To check this, examine one basis simplex at a time: a simplex in the overlap has two copies with their difference in the kernel; one outside the overlap has exactly one copy; simplices in \(A\) are zero. Dualizing and using (D.1) gives relative Mayer–Vietoris, with the cochain restriction difference \((a,b)\mapsto a-b\). The same argument proves the excision needed below: for a finite union of disjoint open discs about a finite subset \(Z\subset Y\), the cover consisting of those discs and \(Y\setminus Z\) identifies the relative small chains for \((Y,Y\setminus Z)\) with the direct sum of the disc-relative complexes.

The two-hemisphere argument of C.4 now applies to integral singular homology and cohomology, using (D.3) and the integral prism. Reduced homology of \(S^0\) is \(\mathbb Z\) in degree zero: the augmentation \(\mathbb Z^2\to\mathbb Z\) is the sum. The contractible hemispheres and their equatorial overlap consequently give, by induction, \(\widetilde H_k(S^m;\mathbb Z)=\mathbb Z\) for \(k=m\) and zero otherwise. Radial retraction and the pair sequence then give

Reflection acts by \(-1\) on the chosen orientation generator: it interchanges the two hemispheres, reversing their difference in (D.3). The positive-determinant path and the decomposition into a reflection and a positive-determinant map in C.4 therefore prove the determinant sign rule integrally as well. This fixes the fiber orientations used below.

D.2. The product maps and their homotopies

We give the product comparison needed for a trivial bundle. On chains the tensor differential is \(\partial(c\otimes d)=\partial c\otimes d+(-1)^{\deg c}c\otimes\partial d\). For \(\sigma:\Delta^n\to X\times Y\), define the front-back map

For a \(p\)-simplex \(c\) and a \(q\)-simplex \(d\), define \(B(c\otimes d)\) as follows. Take all lattice paths \(w\) from \((0,0)\) to \((p,q)\) with horizontal and vertical unit steps. If their successive vertices are \((i_k,j_k)\), let \(\ell_w:\Delta^{p+q}\to\Delta^p\times\Delta^q\) send vertex \(k\) to \((e_{i_k},e_{j_k})\). Set

Here \(w_a\) records whether step \(a\) is vertical or horizontal. The exponent counts ordered pairs of step positions: the word \(VHH\) has two such pairs and therefore sign \(+1\). Interchanging adjacent unlike steps changes this count by exactly one; all pairs involving any other step keep their contribution.

Here are the boundary checks, including the signs. In \(\partial A\sigma\), deletion of a vertex before the split has its usual face sign and deletion after it has the tensor sign \((-1)^p\) followed by its sign in the back face. They match \(A\partial\sigma\). The remaining terms delete the shared vertex from one of the two factors; consecutive splits give identical tensors with opposite signs, and cancel. For \(\partial B\), an interior face obtained by deleting the corner between unlike steps occurs for both orders of those two steps. Their inversion numbers differ by one and their deleted-vertex index agrees, so they cancel. Deleting a vertex between two horizontal steps removes a vertex \(i\) of \(\Delta^p\). If \(j\) vertical steps precede it, the deleted index is \(i+j\), and deleting a horizontal step changes \(\nu\) by \(j\). The resulting sign is thus \((-1)^i\) times the smaller shuffle sign. For two vertical steps the change in \(\nu\) is \(p-i\), so the resulting sign is \((-1)^{p+j}\), as required for the second tensor factor. The initial and final faces obey the same calculation with \(i\) or \(j\) at an endpoint. Thus \(A\) and \(B\) are chain maps.

For completeness, their inverse homotopies can be constructed on universal simplices. Suppose the homotopy for \(1-BA\) is defined below degree \(n\). In \(C_n(\Delta^n\times\Delta^n)\), take the diagonal simplex \(\delta_n\) and the cycle

Its boundary is zero by the lower-degree homotopy identity. In degree zero it is zero; otherwise the cone to a fixed point in the convex space \(\Delta^n\times\Delta^n\) fills it. A cone on an augmented cycle has boundary that cycle by the cone formula in A.3. Choose this filling as the universal value, and push it forward under \(\sigma_X\times\sigma_Y\) to define \(H\sigma\). Face restrictions are pushforwards of their corresponding lower-dimensional universal simplices. Therefore the pushed version of (D.7) gives \(\partial H+H\partial=1-BA\) on every simplex. This construction is natural in \(X,Y\).

The other composition is handled on the universal tensor \(\operatorname{id}_{\Delta^p}\otimes\operatorname{id}_{\Delta^q}\). The augmented chain complex of each simplex contracts to one chosen vertex by coning: if \(K,L\) are the contractions, then \(\partial K+K\partial=1-i\epsilon\) and \(\partial L+L\partial=1-j\epsilon\) on the nonaugmented complexes. The tensor operator

has \(\partial T+T\partial=1-(i\epsilon)\otimes(j\epsilon)\); expanding the tensor differential cancels the two mixed terms. Consequently every positive-degree cycle in this tensor complex bounds. Induct on \(p+q\), fill the cycle \((1-AB)(\operatorname{id}\otimes\operatorname{id})-H'\partial(\operatorname{id}\otimes\operatorname{id})\) with (D.8), and push the filling under \(c\otimes d\). In degree zero the cycle is zero. This proves \(\partial H'+H'\partial=1-AB\).

All these universal constructions stay inside the two simplex images. They preserve \(C_\bullet(X)\otimes C_\bullet(Y_0)\) and \(C_\bullet(X\times Y_0)\) for a subspace \(Y_0\subset Y\). We have therefore proved the relative chain homotopy equivalence

The associated cup operation is equally explicit. For a \(G\)-valued \(p\)-cochain \(a\) and an integer-valued \(q\)-cochain \(b\), put \((a\smile b)(\sigma)=a(\sigma[0,\ldots,p])b(\sigma[p,\ldots,p+q])\). The front-back cancellation just checked gives \(d(a\smile b)=da\smile b+(-1)^pa\smile db\). Restrictions commute with this operation, and evaluating on three successive faces proves associativity. If \(b\) is relative, the product is relative. Right multiplication by a fixed closed \(b\) commutes with the differential without an extra sign. It thus commutes with the connecting homomorphisms of (D.3): extend a cochain, multiply by the restricted global cocycle \(b\), and then take its differential. These identities specify the convention for the Thom and Euler maps.

D.3. The Thom isomorphism on a trivial bundle

We first justify the integral chain splitting used here. A subgroup \(H\) of a free abelian group is free. Indeed, well-order a basis of the ambient group and let \(H_\alpha\) be its intersection with the span of the basis elements before \(\alpha\). At a successor step, projection onto the new coordinate identifies \(H_{\alpha+1}/H_\alpha\) with a subgroup of \(\mathbb Z\), hence either zero or \(n\mathbb Z\). Choose a lift of its generator; this splits the step as a direct sum. At a limit step take the union. Each element has finite support, so the resulting generators form a basis. In particular the cycles and boundaries of a free chain complex are free.

If such a complex \(F_\bullet\) has free homology, choose splittings of \(0\to Z_k\to F_k\to B_{k-1}\to0\) and \(0\to B_k\to Z_k\to H_k\to0\) by lifting bases. In these coordinates the complex is the direct sum of its homology with zero differential and two-term identity complexes on the boundary groups. Mapping each lower member of an identity pair to its upper member contracts those pairs. Apply this to \(F_\bullet=C_\bullet(\mathbb R^r,\mathbb R^r\setminus0)\), using (D.4). It contracts onto \(\mathbb Z\) in degree \(r\). The projection is represented by an integer cocycle \(u_r\), chosen to take value one on the oriented homology generator. It follows at once, by dualizing this contraction, that the relative cohomology of the fiber is \(G\) in degree \(r\) and zero otherwise, for every \(G\).

Tensor this contraction with \(C_\bullet(B)\). If \(h\) contracts the fiber summands, the tensor homotopy is \(c\otimes f\mapsto(-1)^{\deg c}c\otimes hf\); expansion of its boundary gives the claimed tensor contraction. Combine it with (D.9) and dualize with values in \(G\). The resulting cohomology isomorphism is specifically

To verify the formula, the tensor projection sends a tensor of bidegree \((j,r)\) to the value of its second factor under \(u_r\), times the first factor; all other second degrees give zero. Pulling its dual through (D.5) evaluates exactly the displayed front-back cup cochain. This identifies the map, not just the abstract groups. Degrees below \(r\) vanish. In degree \(r\), the integral classes correspond to \(H^0(B;\mathbb Z)\), whose cocycles are the integer-valued functions on points constant along every path. Thus a class is uniquely determined by its restrictions to all fibers, and the class corresponding to the constant function one is their chosen generator.

D.4. Gluing the oriented class on a finite cover

Let \(E\to B\) be an oriented real vector bundle of rank \(r>0\), with a finite cover by oriented trivializations. Put \(E^\times=E\setminus s(B)\), where \(s\) is the zero section. We prove simultaneously the existence and uniqueness of a class

restricting to the positive generator on every fiber, the vanishing in degrees below \(r\), and the isomorphism \(\Phi(a)=\pi^*a\smile u_E\) for every \(G\). The one-chart case is D.3. Suppose the cover has \(N\) charts; let \(U\) be the union of the first \(N-1\), and \(V\) the last. The bundle on \(U\) satisfies the induction hypothesis, and the bundles on \(V\) and \(W=U\cap V\) are trivial. Their classes agree on \(W\): they have the same oriented fiber restrictions, which determine a class there by D.3. Orientation-preserving transition matrices act as \(+1\) by the integral determinant calculation in D.1. Mayer–Vietoris gives a lift to \(H^r(E,E^\times;\mathbb Z)\), unique because \(H^{r-1}(E_W,E_W^\times;\mathbb Z)=0\). The same exact sequence shows vanishing below \(r\).

Choose a global relative cocycle representing this lift. Cup on its right gives a map from the base Mayer–Vietoris sequence with coefficients \(G\) to the relative bundle sequence with degrees increased by \(r\). The squares commute by D.2, including the connecting squares. All vertical maps for \(U,V,W\) are isomorphisms. Here is the exact-sequence argument for the remaining one on \(B\). For surjectivity, restrict a target class to \(U,V\), lift those restrictions through the two isomorphisms, and use injectivity on \(W\) to see their difference is zero. They lift to a base class. The difference from the target then lies in the image of the bundle connecting map; lift its preimage through the isomorphism on \(W\) and add its base connecting image. For injectivity, a base class killed upstairs restricts to zero on \(U,V\), hence comes from a class on \(W\) by the connecting map. Its bundle image is killed by that connecting map, so comes from \(U,V\). Lift this pair back using their isomorphisms and subtract its restriction from the class on \(W\); injectivity on \(W\) makes the remainder zero. Its base connecting image was therefore zero. This proves the required isomorphism in every degree and completes the induction.

Uniqueness also proves naturality under pullback when both bundles have finite trivializing covers: the pulled-back class has the specified fiber generators. Reversing the fiber orientation replaces \(u_E\) by \(-u_E\). This proof uses the finite cover itself and needs no CW replacement or unproved passage to an infinite cover.

D.5. Euler class, Gysin sequence and the dual-line sign

Let \(J:H^r(E,E^\times;\mathbb Z)\to H^r(E;\mathbb Z)\) forget support and define \(e(E)=s^*J(u_E)\). Fiber scaling contracts \(E\) to its zero section, so \(\pi^*\) and \(s^*\) are inverse on cohomology by the prism. In particular \(J(u_E)=\pi^*e(E)\). Associativity and naturality of the cup operation give

Transport the long exact sequence of the pair \((E,E^\times)\) through \(\Phi\) and \(\pi^*\). If the bundle has a continuous fiber norm, its nonzero vectors retract to its unit-sphere bundle \(p:S(E)\to B\) via \(v\mapsto((1-t)+t/\|v\|)v\). We obtain the exact sequence

The map \(p_!\) here means the pair connecting map followed by \(\Phi^{-1}\). This definition fixes all its signs; exactness follows from the proved pair sequence. A complex line has its real rank-two orientation because its transition maps have positive real determinant, as proved in C.4. The Hermitian norm on the tautological line in \(\mathbb C^3\) supplies the required norm in Section 1 (the cone calculation). Hence (D.13) proves its circle-bundle Gysin sequence over both \(\Lambda\) and \(\mathbb Z\).

For a Hermitian line bundle \(L\), take the convention that its metric \(h\) is linear in the first variable. The real bundle isomorphism \(L\to L^*,\ v\mapsto h(-,v)\) is conjugate-linear on each complex line. In an orthonormal coordinate it is complex conjugation, of real determinant \(-1\). Pullback therefore sends \(u_{L^*}\) to \(-u_L\), by their fiber restrictions and uniqueness in D.4. Pulling further along the zero section proves

Thus the minus sign for \(\mathcal O_C(-1)=\mathcal O_C(1)^*\) has a proof independent of the computation of the positive degree.

D.6. Local zeros and the Euler class

There is also a precise local form of the degree computation. Let \(B\) be an oriented surface and \(L\to B\) an oriented real plane bundle with a finite trivializing cover. Let \(t\) be a section with a finite set \(Z=\{x_1,\ldots,x_k\}\) of isolated zeros. It is a map of pairs \((B,B\setminus Z)\to(L,L^\times)\), so \(t^*u_L\) is a relative degree-two class. Its absolute image is \(e(L)\): as maps to \(L\), \(t\) and the zero section are homotopic by fiber scaling.

Take disjoint oriented coordinate discs \(D_i\) on which the bundle is orientedly trivial. The relative excision proved in D.1 gives

Write the section in the \(i\)-th trivialization as a map \(f_i:D_i\to\mathbb R^2\) with its sole zero at \(x_i\). Formula (D.10) identifies the \(i\)-th component of \(t^*u_L\) with \(f_i^*u_2\), an integer multiple \(m_i\) of the oriented local generator. This integer is the local degree. If \(\eta_i\in H^2(B;\mathbb Z)\) is the absolute image of that generator under the inverse of (D.15), we have proved

For a complex coordinate expression \(f(z)=z^m a(z)\), \(m\ge1\), with \(a\) continuous and \(a(0)\ne0\), shrink the disc until \(|a(z)-a(0)|<|a(0)|\). The straight-line homotopy from \(a(z)\) to \(a(0)\) remains nonzero, so it does not change the local degree. Multiplication by \(a(0)\) preserves the real orientation. On a small boundary circle the remaining map is \(z\mapsto z^m\), of degree \(m\). Here is a chain check of that last integer. The loop \(\gamma(t)=\exp(2\pi i t)\) generates integral \(H_1(S^1)\): cutting it into its upper and lower arcs makes the connecting homomorphism for the two-arc cover the difference of its two overlap points, the generator used in D.1. Splitting any path at parameter \(a\) does not change its homology class relative to its endpoints, since the image of the affine triangle with parameter values \(0,a,1\) has boundary equal to the two subpaths minus the original path. Split \(\gamma(mt)\) at \(1/m,\ldots,(m-1)/m\); its pieces are each \(\gamma\). Its homology class is therefore \(m[\gamma]\). The relative connecting isomorphism for the disc identifies this circle degree with \(m_i\). This proves the multiplicity asserted in (D.16) whenever the displayed local factorization has been established.

Equations (D.14) and (D.16) explain the sign and the local contributions to the Euler class in Section 1 (the cone calculation). They do not themselves prove that all positive point classes on the compact curve agree, that a hyperplane section has total multiplicity \(n\), the surface cohomology groups, or the canonical-degree/genus identity. Appendix E supplies those geometric and global topological steps separately. The construction above supplies the classical Thom/Gysin prerequisite used there.

For singular products and the Thom and Gysin constructions, see Hatcher's author text, Chapters 3 and 4.

Appendix E. The compact plane curve, its degree and its genus

We now supply the classical curve inputs in Section 1 (the cone calculation). Let \(F(x,y,z)\) be a nonzero homogeneous polynomial of degree \(n>0\), with no point of \(C=\{F=0\}\subset\mathbb P^2(\mathbb C)\) at which all its first partial derivatives vanish. We prove connectedness, construct a finite decomposition into oriented triangles, compute integral cohomology, identify the positive point class, and calculate the relevant line-bundle degrees. In this classical calculation genus has the topological normalization \(\chi(C)=2-2g\). The proof below gives \(2g=\operatorname{rank}H_1(C;\mathbb Z)=(n-1)(n-2)\), including its integrality. No comparison with coherent or étale cohomology is needed here.

E.1. The analytic and polynomial facts needed for projection

We use analytic functions in the concrete sense of locally convergent complex power series. Their sums, products and compositions are analytic on smaller discs: absolute convergence permits regrouping the series, and their differentiated series converge uniformly on every smaller closed disc. The identity principle follows directly. A series which is not identically zero has the form \(z^m a(z)\), with \(a(0)\ne0\), near a zero; such a zero is isolated. If a function vanishes on a nonempty open part of a connected domain, the set of points where it vanishes in a neighbourhood is open and closed. At a limit point all derivatives vanish by continuity, so its Taylor series is zero there. Thus the function vanishes throughout the domain.

We will need convergence in the implicit function theorem, so we include that step. After translating and scaling, write a convergent series with \(f(0,0)=0\) and \(f_w(0,0)\ne0\) as

Let \(A_r\) be the space of series \(u(t)=\sum u_k t^k\) with norm \(\|u\|_r=\sum |u_k|r^k\). It is a complete normed algebra: multiplication obeys the product bound by the absolutely convergent double sum; a Cauchy sequence converges coefficientwise, and the Cauchy tail estimate, first for finite partial sums and then for their supremum, proves convergence in this norm. Take the closed ball \(u(0)=0,\ \|u\|_r\le s\). For sufficiently small \(s\), and then sufficiently small \(r\), the absolute coefficient sums of \(R\) give

Indeed \(u^j-v^j=\sum_{a=0}^{j-1}u^a(u-v)v^{j-1-a}\), so the first bound follows from \(\sum j|R_{ij}|r^is^{j-1}\le1/2\). All its terms tend to zero since the linear \(w\)-term of \(R\) is absent. The second bound follows by decreasing \(r\). The map \(u\mapsto-at-R(t,u)\) preserves this ball. Its iterates have successive differences bounded by a geometric series and hence converge in \(A_r\) to a unique fixed point \(\phi\). The same estimates on the scalar disc \(|w|\le s\) show that for each \(|t|\le r\) this is the unique small solution. Thus the zero set of \(f\) near the point is the analytic graph \(w=\phi(t)\). Implicit differentiation gives \(\phi'=-f_t/f_w\). The same construction applied to \(h(w)-t\) gives an analytic local inverse whenever \(h'(0)\ne0\).

A nonconstant analytic map between discs has the local form \(t=\zeta^e\), after translating the two centres and changing the source coordinate, for an integer \(e\ge1\). To see this, write \(t(w)=w^e a(w)\), \(a(0)\ne0\). After choosing one \(e\)-th root of \(a(0)\), the binomial series gives an analytic \(e\)-th root of \(a(w)\) on a smaller disc. For precision, the series for \((1+u)^{1/e}\) converges for \(|u|<1\) by the ratio test; its coefficients give \((1+u)P'=P/e\) and \(P(0)=1\), whence \(P^e=1+u\) by comparing coefficients in that differential equation. Set \(\zeta=w a(w)^{1/e}\); its derivative at zero is nonzero, so the inverse just proved applies. In particular such a map is open, and its critical points are isolated unless it is locally constant.

We also need the following growth fact: an entire analytic function bounded by \(M(1+|t|)^k\) is a polynomial of degree at most \(k\). Here is the coefficient argument with its analytic justification. A convergent power series has a local primitive, obtained by integrating its coefficients. Consequently an integral around the boundary of a sufficiently small triangle vanishes. Subdivide a polygonal annulus into triangles small enough to lie in primitive neighbourhoods; internal edges cancel. Applying this to \(a(t)/t^{j+1}\) shows that its integrals over two concentric polygonal boundaries enclosing zero agree. Regular polygons approximate the circles, with both their parametrizations and piecewise derivatives converging, so the circle integrals agree as well. On a small circle, termwise integration of the series gives

For the subdivision assertion it suffices to take concentric regular polygons, join corresponding vertices to form an annular strip of quadrilaterals, divide these into triangles, and repeatedly subdivide. Compactness supplies a Lebesgue number for the primitive neighbourhoods, exactly as in A.3. Thus no global Cauchy theorem has been assumed. Letting \(R\to\infty\) in (E.3) gives \(a_j=0\) for \(j>k\); the identity principle extends this polynomial equality from a neighbourhood of zero to the whole plane.

Finally every nonconstant complex polynomial has a root. Its modulus tends to infinity, so it attains a minimum. If the value at a minimum were \(c\ne0\), expansion there would give \(c+bw^m+O(w^{m+1})\), \(b\ne0\). Choose a ray with \(bw^m/c\) negative real. For sufficiently small \(|w|\), its negative term decreases the modulus, since the remainder is of smaller order, a contradiction. Factoring out one linear factor and repeating proves that a degree-\(n\) polynomial has \(n\) roots counted with multiplicity. It also proves that a nonzero homogeneous polynomial of positive degree in two variables vanishes at some point of \(\mathbb P^1\): use its value at \((t,1)\), and include \([1:0]\) if its degree in \(t\) drops.

E.2. A finite branched projection and connectedness

Projective space is compact: every complex line has a unit representative, so it is a continuous image of the compact unit sphere. It is Hausdorff; for example \(\ell\mapsto vv^*/\|v\|^2\) embeds its set of lines into the Hausdorff space of Hermitian matrices, continuously in each projective chart. The equation \(F=0\) defines a closed subset. At each of its points, a nonzero partial derivative in an affine chart and E.1 give a complex analytic coordinate on \(C\). Such a partial exists: Euler's identity \(xF_x+yF_y+zF_z=nF\), proved termwise on monomials, rules out the possibility that only the omitted homogeneous coordinate has nonzero derivative at a zero. Thus \(C\) is a compact smooth real surface, with the complex orientation justified in C.4.

Choose \(q\notin C\) and coordinates with \(q=[0:0:1]\). Scale \(F\) to have leading coefficient one as a polynomial in \(z\). Projection is then everywhere defined on \(C\):

For fixed \([x:y]\), its fiber consists of the roots of the monic degree-\(n\) polynomial \(F(x,y,z)\). Every fiber is finite and nonempty. In a local analytic coordinate, \(f\) cannot be constant on a neighbourhood, since that would put infinitely many points in one fiber. E.1 therefore gives the local expression \(t=\zeta^{e_p}\). Critical points are locally isolated, including in charts above infinity; their set is closed because the derivative is continuous. Compactness makes it finite. Let \(B\) be the finite set of critical values.

Off \(B\), the map is a finite covering. Indeed each point of a fiber has an inverse chart. Choose disjoint such charts, and shrink the target neighbourhood so that no further points appear: the complement of these charts is compact and its image omits the target point. This produces an evenly covered neighbourhood. Each regular fiber has \(n\) points. In the coordinates \((t,z)\), the kernel of \(df\) on the tangent line is nonzero exactly when \(F_z=0\); a regular fiber thus has simple roots, whose number is \(n\) by E.1. At any value, its local power charts show

This follows by counting a nearby regular fiber in the disjoint power charts; compactness again excludes other inverse images. In particular the branch multiplicities in this argument need not be simple.

We give connectedness its own proof. First a smooth homogeneous plane equation cannot factor as \(F=GH\) with both degrees positive. Suppose their degrees are \(d,e\). As \(G(q)H(q)\ne0\), scale their leading \(z\)-coefficients to make both monic. For each \((x,y)\), consider the square coefficient matrix of

Its determinant \(R(x,y)\) vanishes precisely when the two monic polynomials have a common root. If there is a common factor, take \(u=H/D,\ v=-G/D\) to obtain a nonzero kernel vector. If there is none, factoring into linear factors shows that \(G\mid vH\) implies \(G\mid v\), impossible with \(\deg v<d\) unless \(v=0\); then \(u=0\). Every determinant term is homogeneous of degree \(de\). Explicitly, the entry in output row \(z^k\) and input column \(u_jz^j\) has degree \(d+j-k\), and the corresponding \(v_j\)-entry degree \(e+j-k\). Summing over all columns and subtracting \(\sum_{k=0}^{d+e-1}k\) gives \(de\). If \(R\) is identically zero, any nonzero \((x,y)\) supplies a common root. If it is not, E.1 supplies a projective zero since \(de>0\). At the resulting point \(G=H=0\), all derivatives of \(F=GH\) vanish. This contradicts smoothness and proves irreducibility without assuming a general intersection theorem.

Now suppose \(C=C_1\sqcup C_2\) is a decomposition into two nonempty closed and open subsets. Both are compact. The local power description makes the image of either open, while compactness makes it closed. Since \(\mathbb P^1\) is connected, both images are all of \(\mathbb P^1\). On the complement of \(B\), form the monic polynomial whose roots are the \(z/y\)-coordinates of the points belonging to \(C_1\), with \(t=x/y\). Its degree \(m\) is constant there, with \(0<m<n\); the punctured sphere is path connected because a polygonal path can be detoured around finitely many missing points. Its coefficients are single-valued analytic functions on \(\mathbb C\setminus B\): local inverse sheets are analytic, and continuation only permutes roots within the same closed-and-open subset.

These coefficients extend analytically across a branch value. In a power chart \(t-t_0=\zeta^e\), write the other coordinate as an analytic series \(z/y=h(\zeta)\). All \(e\) nearby roots from this chart belong to the same subset, since the chart is connected. The product \(\prod_{\omega^e=1}(Z-h(\omega\zeta))\) has coefficients given by convergent series invariant under \(\zeta\mapsto\omega\zeta\). Their coefficients with exponents not divisible by \(e\) vanish, so they are convergent power series in \(t-t_0\). Multiplying these local factors proves the extension.

Near infinity use \(s=y/x\) and \(w=z/x\). The monic polynomial \(F(1,s,w)\) has bounded coefficients for small \(s\), and its roots are bounded: if the coefficients after the leading term have modulus at most \(M\), no root has modulus greater than \(1+M\), by comparing its leading term with the geometric sum of the other terms. Since \(z/y=t w\), the coefficient of degree \(k\) in the polynomial for \(C_1\) grows at most as a constant times \((1+|t|)^k\). By (E.3) it is a polynomial of degree at most \(k\). Homogenizing these coefficients gives a homogeneous monic factor of degree \(m\) in \((x,y,z)\). The same reasoning for \(C_2\) gives a factor of degree \(n-m\); their product equals \(F\) on every regular fiber and hence as a polynomial. This contradicts (E.6). Thus \(C\) is connected. It is also path connected: path components of a surface are open by its coordinate discs, and their complements are open as unions of other path components.

E.3. A finite collection of triangles on the actual curve

The projection produces the finite topology model required for integral cohomology. Choose an affine target coordinate with all points of \(B\) finite. Inside that plane take a large triangle containing them. Its exterior in the sphere is a closed disc: radial coordinates from an interior point, followed by inversion of the radial variable, give such a parametrization, with infinity as the disc centre. Joining infinity to the three vertices divides this exterior into three triangles. Together with the inner triangle, this is a four-face decomposition of the sphere, with \(4\) vertices and \(6\) edges. Insert the points of \(B\) one at a time. A point inside a face is joined to its three vertices; a point on an edge splits that edge and the two adjacent faces. In either case \(V-E+T\) is unchanged. We obtain a finite decomposition with all branch values as vertices and \(V-E+T=2\). Near a finite vertex its edges are straight in the affine coordinate. Extra subdivisions may be made without changing this identity.

Over an open edge or open face there are \(n\) disjoint sheets, each mapped homeomorphically onto that cell. Here is the covering argument for the faces. Lift a path by subdividing its parameter interval into inverse-chart intervals; its initial lift determines each succeeding piece uniquely. Subdivide a path homotopy into a grid whose small rectangles map into evenly covered sets. Uniqueness on each rectangle proves that lifts of homotopic paths with the same endpoints have the same endpoint. A disc contracts its paths to radial paths, so lifting paths from its centre gives the disjoint global inverse sheets, continuous by the inverse charts. This establishes the claimed trivialization on the open face without assuming a covering-classification theorem.

The closure of each lifted face is a closed triangle. Along an edge its inverse extends by the inverse chart. At a branch vertex the target face occupies a sector, and the local expression \(t=\zeta^e\) has a single continuous inverse branch on that sector, extending to zero. If the edges are curved in a local parameter, their parametrized sector still has a continuous argument of width less than \(2\pi\), after shrinking; in the finite branch coordinate the edges here are straight. Different vertices of a base triangle remain different points in every lift. These extensions give a continuous bijection from the closed triangle to the lifted closure and hence a homeomorphism, because the domain is compact and \(C\) is Hausdorff. The resulting finite glued collection of triangles maps bijectively to \(C\), including its edges and vertices, and the same compactness argument proves it is the actual topology of \(C\). Fix an ordering of the base vertices and one parametrization of every edge. Choose each face parametrization with those boundary parametrizations: a change of boundary parametrization extends across a disc by the radial formula \(r\theta\mapsto r h(\theta)\). Lift these choices to \(C\). Adjacent singular face simplices then have literally the same edge maps, with the boundary signs dictated by the vertex order; multiplying each face by its orientation sign makes their common edge terms cancel.

Let \(v,e,t\) count its vertices, edges and faces, and put \(R=\sum_p(e_p-1)\). The faces and edges each have \(n\) lifts, and (E.5) counts the vertices above every base vertex. Consequently

The complex orientation orients every face. The two faces adjacent to an edge induce opposite directions on it, as follows by inspecting an oriented coordinate disc across the edge. The adjacency graph of faces is connected. Indeed the faces meeting a vertex are joined successively across edges in its disc neighbourhood. Therefore any union of face-adjacency components, including their edges and vertices, is open and closed in the surface. Connectedness of \(C\) leaves only one component. The graph consisting of its vertices and edges is connected as well: the boundary of every face is connected, so attaching these discs cannot join two different components of that graph.

E.4. Integral cohomology and the positive point class

We verify that the finite chain complex of these triangles computes singular homology; this also fixes its orientation generator. For \(K^0\subset K^1\subset K^2=C\), the relative groups \(H_i(K^j,K^{j-1};\mathbb Z)\) vanish for \(i\ne j\) and are free on the \(j\)-cells for \(i=j\). To prove this at the used dimensions, take an open collar of the boundary of every closed cell, together with \(K^{j-1}\). Pushing radially to the boundary retracts this neighbourhood onto \(K^{j-1}\); continuity follows on each of the finitely many characteristic discs and hence on their quotient. The pair sequence and this retraction replace \(K^{j-1}\) by its neighbourhood without changing relative homology. Cover the rest by slightly larger open cell interiors. Relative small chains from D.1 then split into the finite direct sum of the complexes for an open \(j\)-ball relative to its outer annulus. The ball contracts and the annulus retracts onto \(S^{j-1}\), so (D.4) and the pair sequence give exactly the asserted relative groups.

Their connecting maps give the complex

The first boundary of an oriented edge is its final vertex minus its initial vertex. The boundary of an oriented triangle is its three edges with their induced signs: represent the relative generator by its oriented singular triangle, and use the singular boundary formula. Its boundary loop is the positive circle generator from D.6, so the relative generator agrees with the one used in the ball computation. For \(K^1\), its pair sequence gives \(H_1(K^1)=\ker\partial_1\), \(H_0(K^1)=\operatorname{coker}\partial_1\), and zero higher homology. The pair sequence for \((C,K^1)\) then gives \(H_2(C)=\ker\partial_2\), \(H_1(C)=\ker\partial_1/\operatorname{im}\partial_2\), and the same \(H_0\), with zero groups above degree two. This proves the singular comparison for (E.8).

Since the edge graph is connected, differences of vertices along paths generate the kernel of the sum map \(\mathbb Z^v\to\mathbb Z\), so \(H_0(C)=\mathbb Z\) and \(\partial_1\) has rank \(v-1\). A 2-chain is a cycle precisely when its coefficients on adjacent oriented faces agree: their coefficients at the common edge are their difference, up to the edge's chosen sign. The connected face graph therefore gives \(H_2(C)=\mathbb Z[C]\), where \([C]\) is the sum of all positively oriented faces.

There is no hidden integral torsion in \(H_1(C)\). Choose a spanning tree in the face-adjacency graph, and a root face. Projection of \(\partial_2\) onto the edge coordinates belonging to this tree, restricted to the free group on the nonroot faces, is an isomorphism onto \(\mathbb Z^{t-1}\). To see this, successively remove a leaf other than the root. Its tree edge involves that leaf with coefficient \(\pm1\) and its parent; it solves uniquely for the leaf coefficient, and induction solves all coefficients over the integers. The boundaries of the nonroot faces generate \(\operatorname{im}\partial_2\), since the sum of all face boundaries is zero. Thus projection is a left inverse on this image, which is a direct summand of \(\mathbb Z^e\). Its quotient is free, and \(H_1(C)\), a subgroup of that quotient, is free by D.3. Its rank is

All singular homology groups are therefore free. The splitting of a free chain complex proved in D.3, now applied to \(C_\bullet(C;\mathbb Z)\), yields for every abelian group \(G\)

In degree two the identification is evaluation on \([C]\), independent of the chosen splittings. If \(p\) lies in the interior of a face, the image of \([C]\) in \(H_2(C,C\setminus\{p\};\mathbb Z)\) is its positive local generator: all other face chains vanish relatively, and the remaining face is the oriented local ball class by the same excision. Thus the positive local cohomology class at \(p\) maps to the unique \(\eta\in H^2(C;\mathbb Z)\) with \(\langle\eta,[C]\rangle=1\).

This holds at edges and vertices too, and the needed invariance can be checked explicitly. In an oriented real coordinate disc, take a continuously differentiable cutoff \(\rho\) equal to one near a point and zero outside a smaller compact disc. Such a cutoff can be obtained by a polynomial transition with zero derivatives at both ends, applied to the squared radius. If its Lipschitz constant is \(L\), then \(F_s(x)=x+s v\rho(x)\), \(0\le s\le1\), is a homeomorphism for \(|v|L<1\). Injectivity follows from the Lipschitz inequality; for any target \(y\), iteration of \(x=y-sv\rho(x)\) converges geometrically to its unique inverse image, and the same inequality makes the inverse continuous. Its derivative differs from the identity by norm less than one, so its determinant remains positive along \(s\). Extend it by the identity outside the coordinate disc. It moves the chosen point to a nearby face interior, preserves its local orientation, and is homotopic to the identity on \(C\). Naturality of relative cohomology and of its absolute image shows that the positive class at the original point is also \(\eta\). We have proved equality of all positive point classes, and that this class is an integral generator, before using it to count zeros.

E.5. Hyperplane degree and the canonical line bundle

Recall that \(\mathcal O(-1)\) has fiber the line \(\ell\subset\mathbb C^{m+1}\) over \([\ell]\in\mathbb P^m\); \(\mathcal O(1)\) is its dual, and integral tensor powers define \(\mathcal O(k)\), using duals for negative \(k\). A homogeneous degree-\(k\) polynomial is a section of \(\mathcal O(k)\), since its restriction to \(\ell\) is a homogeneous functional of that degree. This identifies its local zero multiplicities with those of its affine coordinate expressions.

Take a regular value \([a:b]\) of (E.4). The linear section \(bx-ay\) of \(\mathcal O_C(1)\) has exactly \(n\) distinct zeros, one at each point of this fiber. Each zero is simple: its affine expression is a nonzero factor times the target coordinate of the projection minus its value, whose derivative is nonzero at that point. By the local zero formula (D.16) and E.4,

The same section raised to the \(k\)-th power has multiplicity \(k\) at each zero. For \(k>0\), (D.16) gives \(e(\mathcal O_C(k))=kn\eta\). A trivial line has Euler class zero: its nonvanishing section factors through the complement of the zero section, so its pullback of the relative Thom class is zero; it is homotopic to the zero section. For \(k<0\), use (D.14). Thus the formula holds for every integer \(k\), without needing an unproved tensor-additivity theorem.

We next construct adjunction at this exact scope. A nearby line to \(\ell\subset V=\mathbb C^{m+1}\) is the graph of a map from \(\ell\) to a chosen complement. Differentiating graph coordinates identifies the tangent bundle of \(\mathbb P^m\) with \(\operatorname{Hom}(\ell,V/\ell)\); changing the complement gives the same quotient map. Hence there is an exact sequence of complex vector bundles

where the first map identifies \(\operatorname{Hom}(\ell,\ell)\) with scalar multiples of the inclusion. Taking determinants gives \(\det T\mathbb P^m\simeq\mathcal O(m+1)\), after fixing the standard basis of \(V\). The determinant rule used here follows by taking a basis of a subbundle and lifting a basis of its quotient: their wedge is unchanged by changing a lift by a subbundle vector, and changes by the product of the two basis determinants on overlaps.

The differential of the defining section \(F\) along its zero set is a well-defined map \(T\mathbb P^2|_C\to\mathcal O_C(n)\). If the expression of the section changes to \(uF\), its differential along \(C\) changes to \(u\,dF\), because \(F=0\) there; this is exactly the bundle transition. The map is onto by smoothness, and E.1 identifies its kernel with \(TC\). Therefore

Here \(K_C\), also denoted \(\omega_C\) in Section 1 (the cone calculation), is the complex line of holomorphic differentials; the coordinates and changes just used are analytic, so the isomorphism is one of holomorphic line bundles. Degree in this classical calculation is the integer \(\langle e(L),[C]\rangle\). Whenever a holomorphic section has isolated zeros, D.6 proves that this degree is the sum of their multiplicities. In particular (E.11)–(E.13) give \(\deg\mathcal O_C(k)=kn\) and \(\deg K_C=n(n-3)\).

E.6. Ramification, Euler characteristic and genus

The linear projection \(\mathbb C^3\to\mathbb C^2\), \((x,y,z)\mapsto(x,y)\), restricts on each tautological line over \(C\) to a nonzero map onto the tautological line over \(f(p)\). Its kernel line represents \(q\notin C\). These fiber maps give \(f^*\mathcal O_{\mathbb P^1}(1)\simeq\mathcal O_C(1)\). Using (E.12) for \(m=1\), we have \(T\mathbb P^1\simeq\mathcal O(2)\). Consequently the actual derivative of the projection is a section of

In the coordinates \(t=\zeta^{e_p}\) proved in E.1, this section is \(e_p\zeta^{e_p-1}\). Its zeros have multiplicities \(e_p-1\), including zero multiplicity at an unramified point. There are finitely many, and it is nowhere identically zero. The local zero formula, the positive point generator, and (E.11) now give

This includes \(n=1\): the derivative then has no zeros and its bundle is trivial. Combine (E.7), (E.9) and (E.15). The chain computation identifies the alternating number of cells with the alternating rank of singular homology, so it is \(\chi(C)\). We obtain

The product of consecutive integers in (E.16) is even, and it is nonnegative for \(n\ge1\). Thus the topological genus normalization used here gives an integer and the middle rank in (E.10) is \(2g\). Equation (E.13) also proves, rather than assumes at this scope,

In particular a line and a smooth conic have \(g=0\), while a smooth plane cubic has \(g=1\). The proofs give the integral inputs \(H^0(C;\mathbb Z)=H^2(C;\mathbb Z)=\mathbb Z\), \(H^1(C;\mathbb Z)=\mathbb Z^{2g}\), no higher cohomology, and the actual Euler class \(-n\eta\) for its tautological circle bundle. Together with Appendix D these discharge the classical numerical link calculation and the integral extension in Exercise 5. The comparison and vertex attaching complex were proved in Appendix C. Appendices F, H and J supply the general classical adjoint and Whitney-duality arguments. The étale-operation claims require their separate foundations.

For projection, resultants and the topological degree calculation, see the MIT plane-curve notes; for cellular comparison, see Hatcher; for coefficient estimates, see Lebl; for adjunction, see McMullen's notes.

Appendix F. The classical compact-support right adjoint

Here the coefficients are a field \(\Lambda\). Spaces are locally compact Hausdorff and metrizable; the varieties in the classical part have finite atlases by locally closed subsets of Euclidean spaces. We construct the right adjoint of \(R\Gamma_c\), including its local pairing, before using the smooth formula. General constructible biduality and Whitney local triviality are separate assertions. The sheaf resolutions, bounded-below injective comparison and open-closed operations of A.1 and B.1–B.2 are available throughout.

F.1. Compact extension and exact lifting

Write \(F(K)=\Gamma(K,F|_K)\) for a compact subset \(K\), where restriction means inverse image. Two elementary compactness facts will be used. First,

To prove surjectivity, let \(\sigma\in F(K)\). Choose local representatives \(s_i\in F(V_i)\) of \(\sigma\), together with finitely many nonempty compact subsets \(K_i\subset K\cap V_i\) covering \(K\). To obtain these pieces, cover \(K\) by small metric balls whose compact closures lie in representing open sets, take a finite subcover, and intersect their closures with \(K\). For each pair \(i,j\), equality of the germs on \(K_i\cap K_j\) gives an open set \(W_{ij}\subset V_i\cap V_j\) containing that intersection on which \(s_i=s_j\).

Here is the simultaneous shrinking needed for gluing. Write \(N_\epsilon(A)=\{x:d(x,A)<\epsilon\}\). Compactness gives a positive \(\epsilon_i\) with \(N_{\epsilon_i}(K_i)\subset V_i\): otherwise points outside \(V_i\) approaching \(K_i\) would have a subsequence converging to a point of \(K_i\subset V_i\). Likewise, for each pair there is a positive \(\epsilon_{ij}\) such that

Indeed, failure would give \(x_m\notin W_{ij}\) and points \(a_m\in K_i\), \(b_m\in K_j\) with \(d(x_m,a_m)<1/m\) and \(d(x_m,b_m)<1/m\). After a subsequence, compactness gives \(a_m\to a\in K_i\); then \(x_m,b_m\to a\), so \(a\in K_i\cap K_j\subset W_{ij}\), contradicting that \(W_{ij}\) is open. Choose \(\epsilon>0\) smaller than all these finitely many bounds. The restrictions of the \(s_i\) to \(N_\epsilon(K_i)\) now agree on every overlap, so they glue on an open neighborhood of \(K\). Their restriction is \(\sigma\), since the \(K_i\) cover \(K\). If two neighborhood representatives give the same element of \(F(K)\), their germs agree at every point of \(K\); the union of open neighborhoods on which they agree is itself a neighborhood of \(K\). This proves injectivity and hence (F.1). For \(K=\varnothing\), both sides are zero, since the empty open set is allowed in the colimit.

Second, sections on a finite closed cover of a compact set glue when they agree on intersections. Indeed, the space of germs of a sheaf is topologized by the images of its local sections; a section on a subspace is a continuous map to this space over the subspace. The finite closed-set pasting lemma proves continuity of the glued map. That lemma follows by taking inverse images of closed sets and their finite union.

A sheaf \(A\) is c-soft if \(A(X)\to A(K)\) is onto for every compact \(K\). An extension can be chosen with compact support inside any prescribed open neighborhood \(O\) of \(K\). Choose a compact neighborhood \(T\) with \(K\subset\operatorname{int}T\subset T\subset O\). Extend the section prescribed on \(K\) and zero on \(\partial T\), which are disjoint compact sets. The extension vanishes near \(\partial T\), and restriction to \(\operatorname{int}T\), followed by extension by zero, has the required support. Flabby sheaves are c-soft by (F.1). Restrictions of c-soft sheaves to open or closed subspaces are c-soft: compact data there are the same germs on a compact subset of the original space; for an open subspace use the prescribed-support extension just proved.

If \(j:U\hookrightarrow X\) is open and \(A\) is c-soft on \(U\), then \(j_!A\) is c-soft on \(X\). To see this, take a section over a compact \(K\). Its nonzero germs have closed support \(S\) in \(K\), with \(S\subset U\); hence \(S\) is compact. Choose a compact neighborhood \(T\subset U\) of \(S\). On the compact subset \((K\cap T)\cup\partial T\) of \(U\), prescribe the given section on \(K\cap T\) and zero on \(\partial T\). They agree on their intersection, since that intersection avoids \(S\). Extend by c-softness and cut off across \(\partial T\). The resulting compactly supported section in \(U\), extended by zero to \(X\), gives exactly the original germs on \(K\). Closed direct image also preserves c-softness, since the intersection of a compact set with the closed subspace is compact.

Suppose \(0\to A\to B\to C\to0\) is exact and \(A\) is c-soft. A section of \(C\) on a compact \(K\) lifts to \(B(K)\). Choose local lifts on a finite compact cover \(K_1,\ldots,K_r\). Having glued a lift on the first pieces, compare it with the next lift on the compact intersection. Their difference is a section of \(A\) there. Extend that difference using c-softness and subtract it from the next lift. Finite closed gluing continues the induction. For a compactly supported section of \(C(X)\), perform this lifting on a compact neighborhood \(T\) of its support. Near \(\partial T\) the lift is a section of \(A\). Extend its boundary germs by c-softness, subtract, and extend the resulting lift by zero across \(\partial T\). Consequently

is exact. If \(B\) is also c-soft, lifting over a compact set and then extending in \(B\) shows that \(C\) is c-soft. Embed a c-soft sheaf into a Godement injective, iterate this quotient assertion, and apply (F.2). This proves that c-soft sheaves are \(\Gamma_c\)-acyclic, on every open subspace as well. The usual comparison with an injective resolution therefore computes \(R\Gamma_c\) on c-soft resolutions. For bounded-below complexes of acyclic sheaves the same assertion follows by resolving their terms and filtering the resulting double complex: each total degree has finitely many terms above its lower bounds, and the augmented acyclic rows give the comparison quasi-isomorphism.

Both \(j_!\) and closed direct image \(i_*\) are exact, preserve c-soft sheaves, and satisfy the equality of compact-section functors. It follows, by those resolutions, that

This proves the derived compact-support identities rather than assuming a general proper-image theorem. It also gives a useful converse: if \(H_c^1(U,A)=0\) for every open \(U\), then \(A\) is c-soft. Apply the compact-support sequence to \(0\to j_!A|_{X\setminus K}\to A\to i_*A|_K\to0\) for a compact \(K\). By (F.3) the obstruction to extending \(A(K)\) by a compact section is the asserted zero group.

F.2. Filtered compact sections and a finite dimension bound

For a filtered system \(F_\alpha\), sections on a compact \(K\) commute with its sheaf colimit. A section in the colimit has local representatives at some indices; a finite compact cover chooses finitely many indices. On each compact pairwise intersection, equality in the colimit holds at a single later index after another finite subcover. A common upper index thus makes the representatives agree; finite closed gluing gives a section on \(K\). The same argument for zero germs proves injectivity. For compactly supported sections, choose a compact neighborhood \(T\) of the support, lift over \(T\), and pass to a later index where its boundary restriction is zero. Equation (F.1) and extension by zero then give

A filtered colimit of c-soft sheaves is c-soft: represent compact data at one index and extend at that index. Use the functorial Godement resolutions of the system. Their colimit is exact, since this is tested on stalks and filtered colimits of vector spaces are exact. Its terms are c-soft. Apply (F.4) termwise and the acyclicity proved above. This proves (F.4) also for every \(H_c^q\). In particular, arbitrary direct sums of c-soft sheaves are c-soft, and \(\Gamma_c\) commutes with arbitrary direct sums; alternatively, compact data in a direct sum use only finitely many summands, as follows from a finite subcover of local representatives.

For a decreasing filtered family of closed subsets \(Z_\alpha\) with intersection \(Z\), the natural map \(\varinjlim i_{\alpha*}F|_{Z_\alpha}\to i_*F|_Z\) is an isomorphism. At a point of \(Z\) all its stalks are \(F_x\); outside \(Z\) a later closed subset misses the point. The preceding result and (F.3) imply

Say that \(X\) has compact cohomological dimension at most \(N\) if \(H_c^q(X,F)=0\) for every sheaf \(F\) and \(q>N\). Formula (F.3) makes this bound pass to open and closed subspaces. We prove

Let \(q>N+1\) and suppose \(\gamma\ne0\) in \(H_c^q(X\times\mathbb R,F)\). Among closed subsets \(A\subset\mathbb R\) on which the restriction of \(\gamma\) is nonzero, choose one minimal under inclusion. It exists by Zorn's lemma: the intersection of a decreasing chain still has nonzero restriction by (F.5), since a class in a filtered colimit is zero only if it becomes zero at some stage. Such an \(A\) is neither empty nor a singleton, since the corresponding fibers have dimension at most \(N\). Choose \(t\) strictly between two of its points and split it into the two proper closed subsets \(A_-\) and \(A_+\) on the two sides of \(t\). The exact sheaf sequence for this closed cover, checked by its diagonal and difference maps on stalks, gives

The left group vanishes because the intersection is empty or one fiber, and the image of \(\gamma\) on the right vanishes by minimality. This contradicts its nonzero restriction to \(A\). Thus (F.6) holds. A point has dimension zero, so \(\dim_c\mathbb R^m\leq m\).

The bound is also local when it is uniform. For clarity, first prove local c-softness implies c-softness. Given compact data on \(K\), take a compact neighborhood \(T\) and finitely many compact pieces covering \(T\), each lying in an open set on which the sheaf is c-soft. Prescribe the data on \(K\) and zero on \(\partial T\). Extend successively over the pieces: their intersection with the already prescribed compact set is compact in the relevant open set, so c-softness there and finite closed gluing apply. The extension over \(T\) is zero near its boundary by (F.1); extending by zero finishes the proof.

Now suppose an open cover has dimension at most \(N\). For \(N\geq1\), truncate an injective resolution of any \(F\) into the exact sequence

On every open subset of a covering set, dimension shifting gives \(H_c^1(Q)=H_c^{N+1}(F)=0\). The converse in F.1 shows that \(Q\) is locally c-soft, hence c-soft. Sequence (F.7) is a c-soft resolution of length \(N\), proving the global bound. For \(N=0\), the same converse applies directly to \(F\). Thus any space with a finite atlas of locally closed subsets of \(\mathbb R^m\), with bounded \(m\), has finite compact cohomological dimension. In particular, a chosen finite affine atlas of a complex algebraic variety, embedded in finite-dimensional complex affine spaces, supplies the required bound. No stratification or Whitney triviality theorem is used for this bound.

F.3. An exact functor with an explicit sheaf adjoint

Fix such a bound \(N\) and a c-soft sheaf \(M\). We first prove that \(F\otimes M\) is c-soft for every sheaf \(F\). There is a surjection onto \(F\) from the sum of the sheaves \(\Lambda_U=j_!\Lambda\), indexed by pairs consisting of an open set \(U\) and a section of \(F(U)\). The section supplies the map, and these maps are onto on stalks. Repeat for the successive kernels to obtain a left resolution. Tensoring is exact over a field, and every term after tensoring with \(M\) is a sum of \(j_!(M|_U)\), hence c-soft. If \(N\geq1\), take \(N\) consecutive terms of this resolution. On any open subset, their short exact sequences identify \(H_c^1(F\otimes M)\) with \(H_c^{N+1}\) of the leftmost kernel, which vanishes. If \(N=0\), every sheaf is already c-soft. The converse in F.1 proves the claim.

Consequently

is exact by (F.2), and preserves direct sums by (F.4). It therefore preserves cokernels and all colimits, since a colimit is the cokernel of the two maps between the sums indexed by the arrows and objects of its diagram. Its right adjoint on a vector space \(V\) has the explicit form

Restrictions are precomposition with extension by zero. To check the sheaf condition, for an open cover of \(U\) use the exact presentation \(\bigoplus_{\alpha,\beta}\Lambda_{U_\alpha\cap U_\beta}\to \bigoplus_\alpha\Lambda_{U_\alpha}\to\Lambda_U\to0\). Its exactness is checked on stalks: all copies at a point are identified by their pairwise differences. Apply \(T_M\) and then \(\operatorname{Hom}(-,V)\); the resulting kernel diagram is precisely the sheaf gluing diagram for (F.9).

Moreover,

For \(F=\Lambda_U\) this is (F.9), using the exact open adjunction. For a sum of such sheaves it follows by products. For general \(F\), take the surjection by sections above, and a similar surjection onto its kernel. Both sides of (F.10) are the same kernel for that two-term presentation. The identification is natural because its value on every map \(\Lambda_U\to F\) is the prescribed section map. This presentation also includes the linear relations between sections; a diagram containing only restrictions of individual sections would not suffice. Since \(T_M\) is exact and every vector space is injective, (F.10) shows that \(\mathcal D_M(V)\) is an injective sheaf.

F.4. Constructing the right adjoint, with its differential

Apply (F.7) to \(\Lambda_X\). It gives a c-soft resolution \(\Lambda_X\to L^\bullet\) supported in degrees \(0,\ldots,N\); for \(N=0\) use \(L^0=\Lambda_X\). All its terms are flat, since the coefficients form a field. For a bounded-below complex \(F^\bullet\), the augmentation \(F^\bullet\to F^\bullet\otimes L^\bullet\) is a quasi-isomorphism: on stalks the bounded exact augmented complex of vector spaces splits and contracts. Its total terms are finite sums of the c-soft sheaves of F.3. Hence

For a bounded-below complex \(V^\bullet\) of vector spaces define

Here the second term uses contravariance in \(L\), and out-of-range components are zero. Squaring the differential gives zero: the two mixed terms have opposite signs and the two square terms vanish. The complex is bounded below by the lower bound of \(V\) minus \(N\), and every term is injective by F.3. B.1 therefore makes it a valid derived-Hom target.

Currying (F.10) gives an isomorphism of mapping complexes

To check its signs, let a homogeneous map have degree \(n\) and restrict to \(F^i\otimes L^p\). On both sides its differential has the three terms \(d_Vh_{i,p}-(-1)^n h_{i+1,p}d_F-(-1)^{n+i}h_{i,p+1}d_L\). Thus this is a complex isomorphism, not only a degreewise bijection. Products in a mapping complex cause no rearrangement issue: the \(p\)-index has only finitely many values. Complexes of vector spaces compute derived Hom, since every acyclic vector-space complex splits and contracts; the same bounded-below injective assertion on \(X\) was proved in B.1. Equations (F.11)–(F.13) therefore prove

Mapping complexes here are allowed to be unbounded; no bounded-below assertion about \(R\operatorname{Hom}\) is needed. Taking degree zero gives the desired adjunction on bounded-below derived categories. Formula (F.12) is functorial, and sends quasi-isomorphisms of \(V\) to isomorphisms in the derived category, either by (F.14) and Yoneda or directly by the injective comparison. Different choices of \(L\) give the unique compatible right adjoint by the same Hom characterization. This constructs \(a^!\), rather than assuming its existence.

F.5. Local pairing, orientation, and smooth local systems

Put \(\omega_X=a^!\Lambda\). Restriction of the construction to an open \(U\) uses the c-soft resolution \(L|_U\) and gives \(\omega_X|_U=\omega_U\). Since (F.12) consists of injectives, its sections over \(U\) calculate derived sections. Formula (F.9) consequently gives the actual local comparison

More generally, for a bounded complex \(F\), internal Hom into this injective complex and (F.14) on \(U\) give

Here internal derived Hom can indeed be computed this way: restriction of an injective sheaf to an open set is injective, because its left adjoint \(j_!\) is exact. Thus sheaf Hom into an injective is exact in its first variable. Moreover \(\mathcal Hom(F^i,I)\) is injective when \(I\) is injective: its Hom functor is \(\operatorname{Hom}(-\otimes F^i,I)\), which is exact over a field. The bounded complex \(F\) therefore gives a bounded-below complex of injectives by finite totalization; its ordinary sections compute the required derived sections. The maps in (F.15)–(F.16) are compatible with shrinking \(U\), where their right sides use the dual of extension by zero. Evaluation followed by the adjunction counit \(R\Gamma_c(\omega_X)\to\Lambda\) is the compact-support pairing: under (F.14), that counit corresponds to the identity of \(\omega_X\), so (F.16) identifies this pairing, including its evaluation map, with the displayed derived linear dual.

On a real \(m\)-manifold, compute stalks of (F.15) on coordinate balls. By C.5 their only compact-support group is \(\Lambda\) in degree \(m\). Extension by zero from a smaller ball preserves its local oriented generator: represent it with support on a still smaller closed ball and use the support excision of C.5; the two representatives then have the same relative sphere generator. Hence the sole cohomology sheaf of \(\omega_X\) is in degree \(-m\), with transition signs the orientation signs of C.4. The dual sign is the same sign. If \(\mathrm{or}_X\) denotes this rank-one orientation sheaf, the truncation construction in B.1 gives

Complex charts have positive real determinant by C.4, so their positive generators agree and \(\mathrm{or}_X=\Lambda_X\) canonically for a complex \(d\)-manifold. A finite-rank local system \(L\) is locally a finite sum of copies of \(\Lambda\). Sheaf Hom out of it is exact and is \(L^\vee\otimes-\), as follows on these trivializations, whose changes of basis act by inverse transpose. Therefore

Finite-dimensional evaluation gives biduality for these local systems and their shifts. This proves the classical assertion of Proposition 2.3. It does not yet prove biduality for all constructible complexes across singular strata.

F.6. Closed support and the costalk of a dual

Let \(i:Z\hookrightarrow X\) be closed, with the same finite-dimensional hypotheses. B.2 constructs \(i^!\) as the derived right adjoint of the exact \(i_*\). By (F.3), \(R\Gamma_c(X,i_*H)=R\Gamma_c(Z,H)\). Applying the two adjunctions, for every bounded-below \(H\) we obtain \(R\operatorname{Hom}_Z(H,i^!\omega_X)=R\operatorname{Hom}_\Lambda(R\Gamma_c(Z,H),\Lambda)\). Yoneda and F.4 identify

This also proves the independence of a closed ambient embedding in the definition of the structural dualizing complex. Its counit is the composed counit, since both correspond to the identity under the composed adjunction.

For bounded \(K\), tensor-Hom adjunction and the stalkwise identity \(i_*H\otimes K=i_*(H\otimes i^*K)\) now give

Explicitly, mapping a bounded-below \(H\) into the left side gives the successive complexes \(R\operatorname{Hom}_X(i_*H,R\mathcal Hom(K,\omega_X))\), \(R\operatorname{Hom}_X(i_*H\otimes K,\omega_X)\), and \(R\operatorname{Hom}_Z(H\otimes i^*K,\omega_Z)\), which identify with maps into the right side. The tensor-Hom comparison follows by ordinary currying against the injective resolution used in F.4; tensoring over a field preserves acyclic complexes, as is checked on stalks. These identifications commute with evaluation, so Yoneda supplies the stated natural map and its inverse. For bounded \(H\), the same calculation on every open set gives \(D_Xi_*H=i_*D_ZH\). Likewise the open adjunction and \(j_!H\otimes K=j_!(H\otimes j^*K)\) give \(D_Xj_!H=Rj_*D_UH\).

For \(Z=\{x\}\), the structural adjoint is the identity on vector spaces. Equation (F.20) yields the second formula in (2.1), \(H^m(i_x^!D_XK)=H^{-m}(i_x^*K)^\vee\), without assuming constructible biduality. The first formula in (2.1) and constructible biduality are subsequently proved in Appendix J from the Whitney geometry of I and the finite-neighborhood criterion of G.1. Appendix H supplies the general classical operation identities, and K.8 proves their algebraic constructibility hypotheses.

For compact-support duality and soft resolutions, see Mathew, §§3–5, and Schnürer–Soergel, §§5–7.

Appendix G. Biduality from finite neighborhoods, and the actual cone models

We continue over a field on the finite-dimensional classical spaces of Appendix F. For a complex of vector spaces write \(P^\vee=R\operatorname{Hom}_\Lambda(P,\Lambda)\). A bounded complex with finite-dimensional cohomology is perfect: choose complements to its cycles and boundaries, split off the resulting contractible pairs, and retain its finite-dimensional cohomology complex. Its evaluation into \(P^{\vee\vee}\) is an isomorphism. For homogeneous \(v\in P^r\) it is the map \(\phi\mapsto(-1)^r\phi(v)\); this sign makes evaluation a chain map with our Hom differential.

G.1. A local biduality criterion with its actual comparison maps

Proposition. Let \(F\) be bounded. Suppose that a point \(x\) has a cofinal basis of open neighborhoods \(U\) for which \(R\Gamma(U,F)\) is perfect. Then the canonical evaluation \(F\to D_XD_XF\) is an isomorphism on the cohomology stalks at \(x\). If this holds at every point, it is an isomorphism of sheaf complexes.

First note that all duals in this assertion exist and are bounded. F.4 gives \(\omega_X\) as a bounded complex of injectives, in degrees \([-N,0]\). A bounded representative of \(F\), and the exact tensor–Hom argument in F.5, therefore computes its internal dual by a bounded complex of injectives. The same applies to the second dual.

Choose nested neighborhoods from the stated basis and compact neighborhoods with \(U_{n+1}\subset K_n\subset U_n\), all containing \(x\). They can be made cofinal by requiring \(U_n\) to lie inside the metric ball of radius \(1/n\). Put

Extension of supports gives natural maps

Their composites are the original support-extension maps. These can be represented on the same injective resolution, using the actual inclusions of support-section complexes and the filtered unions over compact supports. After applying a contravariant functor and then taking filtered colimits of cohomology, the two resulting systems have the same colimit: the displayed interleaving supplies maps both ways, and their composites are transitions, hence identities in the colimit. This elementary argument does not assert that duality commutes with an inverse limit.

F.6 supplies, for the closed inclusion of a compact \(K\),

The second equality uses the compactness of \(K\), so that its ordinary and compact-support sections agree, and the structural adjunction F.14 on \(K\). Closed subsets inherit the required finite dimension by F.2.

On the other hand, ordinary restriction interleaves the systems \(R\Gamma(U_n,F)\) and \(R\Gamma(K_n,F|_{K_n})\): restrict first from \(U_n\) to \(K_n\), then to \(U_{n+1}\). The maps here are the derived restriction maps induced by the sheaf adjunction units. Their composites are the original restrictions. Interleaving still holds after applying the covariant double-dual functor. Apply F.16 to \(D_XF\), take stalks, then use (G.1), (G.2), and this ordinary interleaving. For every \(q\) the result is

We must check the morphism, not only the groups. The pairing giving the first line is evaluation followed by the counit of F.14. Under closed adjunction, restricting a section of \(F\) to \(K_n\) pairs it with the supported dual section in (G.2). Currying this pairing gives precisely the vector-complex evaluation \(v\mapsto(\phi\mapsto(-1)^{|v|}\phi(v))\). Thus the map from \(F_x\) in (G.3) is the filtered colimit of the evaluation maps for the ordinary section complexes. Each is an isomorphism by the assumed perfectness. Finally, on an injective resolution, exact filtered colimits give \(\varinjlim_n H^q(R\Gamma(U_n,F))=H^q(F_x)\). This proves the proposition, including its canonical evaluation map. No stabilization or finiteness of the intermediate compact-neighborhood groups was assumed. Those groups were handled by the actual interleaving maps.

G.2. Local coefficients and finite cohomology of the links used here

We record the local-coefficient extension of C.1–C.2 needed below. Let \(L\) be a local system on a locally compact metrizable manifold. Transport along paths and its invariance under endpoint-fixed homotopies were proved in Appendix A.4. A degree-\(n\) singular cochain assigns to a simplex \(\sigma\) an element of \(L_{\sigma(v_0)}\). Its differential on an \((n+1)\)-simplex is

where \(\tau_{10}\) is transport along the first edge from \(v_1\) to \(v_0\). Every codimension-two face in \(d^2\) occurs twice with opposite signs. The only apparently different transports are the direct edge from \(v_2\) to \(v_0\) and the two consecutive edges; they agree by the homotopy inside the simplex. Thus \(d^2=0\).

The small-chain, subdivision and prism constructions of C.1 extend to these cochains as follows. Pull the local system back to each simplex or prism, and trivialize it by transport from its first vertex. That domain is contractible, so transport is path independent there. Each subdivided simplex receives the transport of its first-vertex value along the straight edge in the original domain. On common faces these identifications agree, and the cancellations proving the subdivision and prism identities in C.1 are consequently unchanged. This proves both the small-cochain comparison and homotopy invariance, with the induced transport on coefficients.

We first justify the countability needed to apply C.2. Every connected component of a metrizable manifold has a countable basis. Here is a metric proof, allowing arbitrarily many components. A separable metric space has a countable basis, given by rational-radius balls about a countable dense set; its subspaces consequently have countable bases and are separable, by choosing a point in each nonempty basic open set. Let \(X_n\) consist of the points whose radius-\(1/n\) ball is separable. Coordinate charts show that the \(X_n\) cover the manifold. Starting with \(A_0=\{x_0\}\), define

Each \(A_k\) is separable: replace each set of centers \(A_k\cap X_n\) by a countable dense subset, which gives exactly the same union of open balls of that fixed radius. Those balls are separable by definition of \(X_n\). A countable union of separable subspaces is separable, by taking the union of their countable dense sets. The set \(A\) is open because every point belongs to some \(A_k\cap X_n\). It is also closed. If \(x\in\overline A\), choose \(n\) with \(x\in X_n\) and \(y\in A\) with \(d(x,y)<1/(4n)\). Then \(B(y,1/(2n))\subset B(x,1/n)\) is separable, so \(y\in X_{2n}\). The construction includes \(B(y,1/(4n))\) in \(A\), and this ball contains \(x\). Thus the connected component of \(x_0\) lies in the separable set \(A\) and inherits a countable basis. Components are open, since coordinate balls are path connected. The same argument applies to every open submanifold. Consequently C.2's compact exhaustions and locally finite closed refinements can be chosen separately on its components; their union remains locally finite because the components are disjoint and open.

Sheafify the cochains. On a trivializing convex chart this is the exact augmented cochain resolution proved in A.2–A.3. Hence it resolves \(L\) on stalks. It is flabby by the representative argument of C.2, now with the componentwise refinements just justified: refine an open cover so that the representatives lie in trivializing charts, extend their values on the small simplices to the ambient space by zero, and sheafify. A simplex lies in one component, so the componentwise representatives give one cochain on the entire open set. Agreement is a local assertion on smaller simplices and is preserved by these extensions. Thus its ordinary sections compute sheaf cohomology. The refinement comparison of C.2 identifies them with (G.4); the subdivision identities just checked prove that comparison for these local coefficients. In particular, a product with an interval induces the ordinary-cohomology equivalence given by projection and a slice, with all interval restrictions compatible.

We also need this interval assertion for a bounded complex \(J\) whose cohomology sheaves are finite-rank local systems on \(M\times I\). Let \(p:M\times I\to M\). For a sufficiently small contractible chart \(V\subset M\), each cohomology sheaf of \(J\) is constant on \(V\times I\). The preceding cochain contraction computes its sections in degree zero. Apply the finitely many truncation triangles of B.1 to \(J\); they show that \(R\Gamma(V\times I,J)\) has the cohomology of its stalk and that restriction to a slice is an isomorphism on those groups. Since ordinary direct image preserves injectives, these computations on \(V\) identify the counit

as a quasi-isomorphism on every stalk. Put \(J_0=Rp_*J\); it is bounded with finite-rank locally constant cohomology and is identified with every slice restriction. Derived composition of ordinary sections and direct image gives \(R\Gamma(M\times I,J)=R\Gamma(M,J_0)\). These identifications come from the counit, so restriction to a subinterval yields the same comparison, rather than an unrelated isomorphism of vector spaces.

For a circle, A.4 identifies the ordinary cohomology of a finite-rank local system with \([V\xrightarrow{T-1}V]\). Hence it is perfect, as is a finite disjoint union of circles. The same is true of every finite-rank local system on the circle bundle \(M\to C\) in Section 1 (the cone calculation). Here are the finite-cover details. The actual finite triangle decomposition of \(C\) was constructed in E.3. Subdivide it until each open vertex star lies in a trivializing chart for the tautological line bundle. This is possible: the compact realization has a Lebesgue number for the finite trivializing cover, subdivision decreases the simplex mesh by the factor at most \(2/3\), and a star has diameter at most twice that mesh. The homeomorphism from the compact triangle realization to \(C\) is uniformly continuous. The Lebesgue-number assertion itself follows by choosing, at every point, a metric ball contained in a covering set, taking a finite subcover by the half-radius balls, and taking the minimum of their radii.

If the original triangle decomposition has more than one cell with the same vertex set, first use its barycentric subdivision: vertices then label cells and simplices label chains of faces, giving an ordinary simplicial complex with the same realization. Every nonempty finite intersection of its open vertex stars is the open star of their spanned simplex. On this triangulated surface it is an open disc: a triangle has its interior, an edge has its two adjacent triangle halves, and a vertex has the cyclic fan of triangles given by the local disc model in E.3. Each lies in a line-bundle chart. The inverse image of such an intersection in \(M\) is therefore a product of an open disc with a circle. By (G.4) and interval or disc contraction, the cohomology of a finite-rank local system there is the finite two-term circle cohomology.

Finally use the derived two-open-set sequence proved by the sheaf resolution in A.4. Finite induction shows that a finite cover whose nonempty intersections have perfect section complexes has perfect global sections: separate the last set from the union of the preceding ones, apply the induction to their intersections, and take the fiber of the two-set difference map. Kernels, cokernels and extensions of finite-dimensional vector spaces remain finite-dimensional, and the finite number of these triangles keeps the degree range bounded. This proves perfectness on \(M\). Applying finitely many truncation triangles proves it for every bounded \(J_0\) with finite-rank locally constant cohomology on this link.

G.3. The attaching complex on an actual open cone

Let \(M\) be one of the compact links just considered, and let \(U_R=(M\times[0,R))/(M\times\{0\})\) be the open cone, with vertex \(v\). Write \(j:U_R\setminus\{v\}\hookrightarrow U_R\) and \(i:\{v\}\hookrightarrow U_R\). Suppose \(F\) is bounded, its vertex stalk \(A=i^*F\) is perfect, and its cohomology on the puncture consists of finite-rank local systems. Equation (G.5) writes \(j^*F=p^*J_0\). Put

Here \(B\) is perfect by G.2. The identification \(i^*Rj_*j^*F=B\) uses the compatible subinterval restrictions in (G.5) and the injective stalk calculation of A.4. The map \(a\) is the actual restriction of the adjunction unit \(F\to Rj_*j^*F\).

For completeness, the reconstruction required here follows directly from B.2:

The units give maps from \(F\) to the two summands, whose compositions agree by naturality. Represent the units on complexes and choose the resulting commuting homotopy; the cone differential of B.1 then gives the map into the displayed fiber. On the puncture it is the identity of \(j^*F\). At the vertex the fiber is \(\operatorname{Fib}(B\oplus A\to B)\), and projection to \(A\) is a quasi-isomorphism, since the \(B\xrightarrow1B\) pair contracts. The map from \(F\) induces the identity under this projection. It is therefore a quasi-isomorphism on every stalk, proving (G.7). This also shows that the attaching map, rather than merely its effect on cohomology dimensions, is being retained.

Taking ordinary sections in (G.7) gives the same contractible pair and hence

The restriction from the first complex to the second is \(a\). All smaller-radius comparisons use the original units and the product restrictions in (G.5), so (G.8) is compatible with those restrictions.

Let \(K_r\) be the closed subcone of radius \(r<R\), which is compact. The support fiber formula of C.5, now for \(F\), gives

Increasing \(r\) restricts the complementary annulus and induces the identity on \(B\) under (G.5); it induces the identity on \(A\) as well. These support groups consequently have identity transition maps in the displayed model. The sets \(K_r\), \(r\uparrow R\), are cofinal among compact subsets of \(U_R\). Taking their filtered union in an injective resolution, as in C.12, proves

The last equality is B.2's point-support fiber. Its comparison with (G.9) is the support-extension map: restriction from the puncture to an annulus gives the identity of \(B\). Thus extension from \(U_r\) to \(U_R\) also induces the identity on (G.10). We have proved the local stabilization with its actual maps; no general stratified homotopy theorem has been presumed.

Ordinary neighborhoods at the vertex have the perfect section complex \(A\). At every other point, a sufficiently small chart trivializes the finite local cohomology systems, and the truncation argument of G.2 gives perfect sections there. G.1 proves biduality for \(F\). F.16 and (G.10), with their compatible extension maps, give the other point exchange, and F.20 gives its companion:

Both are perfect. Away from the vertex, F.5 and the local truncation argument show that the dual still has finite-rank locally constant cohomology. Thus duality preserves these two-stratum constructible complexes, and G.1 applies to their duals too. In particular, F.6's identity \(D_Xj_!H=Rj_*D_UH\), followed by this actual biduality, gives the reverse identity \(D_XRj_*H=j_!D_UH\) for these models. This conclusion has been proved at their full bounded-complex scope, not only for constant sheaves.

G.4. Consequences for the curve and cone calculations

A disc with its centre as vertex is a cone on a circle, and the model of finitely many discs with their centers identified is a cone on the finite disjoint union of those circles. The radial maps and quotient topology were proved in B.3–B.4. A finite-rank local system on each punctured branch has the two-term complex of A.4. Thus G.3 proves biduality and both stalk/costalk exchanges for every bounded complex constructible for these branch-and-vertex decompositions. This gives both classical punctured-line duality formulas of Section 2.

For the branch-normalization sheaf \(N\) of Section 1 (the node calculation), one also obtains \(D_XN=N[2]\) canonically. Locally \(N\) is the sum of the closed direct images of the constant sheaves on the separate smooth discs, by B.4. F.6 and F.5 dualize each such summand to the same summand shifted by two. The local degree \(-2\) generators are the positive complex orientation generators on the respective branches. They agree when a branch neighborhood is shrunk, by C.5, and are permuted when branch labels are permuted. They therefore identify the sole cohomology sheaf of \(D_XN\) with \(N\). The truncation isomorphism gives \(D_XN=N[2]\), also across overlaps with the smooth part of the curve. Dualizing the actual short exact sequence (1.2) now proves the triangle (2.2). This proves that classical dual object and its attaching maps at the branch model. L.2–L.4 supply the general algebraic node criterion and identify its actual normalization with these analytic branches, so the calculation applies to every complex algebraic node.

For the affine cone of Section 1 (the cone calculation), C.3 proves that the neighborhoods really are the cones \(U_R\) on its compact circle-bundle link. G.2 proves the needed link finiteness even with finite-rank local coefficients. Therefore G.3 proves both dual exchanges and biduality for every bounded complex constructible for its smooth puncture and vertex. For \(F=\Lambda_X\), \(A=\Lambda\), \(B=C^\bullet(M;\Lambda)\), and \(a\) is the constant-cochain map of C.3. Equations (1.3), (G.10) and (G.11) prove all the asserted classical dual stalk and costalk groups in Proposition 1.5, including their degree signs. For \(F=Rj_*\Lambda\), the map \(a\) is the identity of \(B\), so its costalk and the stalk of its dual vanish, as the reverse extension identity also shows.

G.1 isolates the exact finiteness input for general biduality, while G.2–G.4 verify it on the actual curve and cone spaces used in Sections 1–2. Appendices I–J subsequently construct the general normal neighborhoods and prove coefficient stabilization, compact-link finiteness and fixed-stratification duality. K.6 subsequently proves existence of finite algebraic Whitney refinements.

For the local model and duality, see Goresky, §§9.1–9.4 and 12.4–12.8. General Whitney neighborhoods and their constructible coefficients are treated in Appendices I–J.

Appendix H. Classical proper-support operations and their duality identities

Let \(f:X\to Y\) be a continuous map between locally compact Hausdorff metrizable spaces of the finite compact cohomological dimension proved in F.2. Coefficients are a field. All derived inputs are bounded below unless a stronger bound is stated. The same hypotheses are imposed on spaces in a base-change square; they hold for the classical algebraic fiber products in this course. This appendix proves the operation identities used in Sections 2–3. Preservation of algebraic constructibility requires the additional geometric arguments subsequently proved in K.6–K.8.

H.1. Proper supports and the fiber map

We call a map proper when the inverse image of every compact subset is compact. For maps between locally compact Hausdorff spaces this condition implies that the map is closed. Indeed, to test whether a point \(y\) lies in the image of a closed set, choose a compact neighborhood \(T\) of \(y\). Its inverse image is compact, and the image of its intersection with the closed set is compact, hence closed in \(T\). If \(y\) is a limit of the original image, this compact image contains \(y\). Properness is local on the target: a compact set has a finite closed shrinking of an open cover by relatively compact neighborhoods, and its inverse image is then a finite union of compact inverse images. It is preserved under base change: over a compact subset of the new target, the pullback is a closed subset of the product of that compact set and the compact inverse image of its image in the old target. Closed restrictions and finite unions of proper supports remain proper for the same reasons.

Define

The preceding local property proves the sheaf axiom. Finite unions of supports prove closure under addition. The functor is left exact: a section in the kernel of a sheaf map lifts uniquely into its kernel sheaf, with exactly the same support. If \(f\) is proper then \(f_!=f_*\). For the structural map this is \(\Gamma_c\), and for open or closed inclusions it agrees with the functors of B.2.

The ordinary inverse image \(f^*\) is sheafification of the usual neighborhood-colimit presheaf. Its stalk at \(x\) is \(F_{f(x)}\), so it is exact. The section definition of \(f_*\) gives its adjunction with \(f^*\); hence \(f_*\) preserves injectives. Injective resolutions therefore give \(Rf_*\) and the derived ordinary adjunction, including composition of ordinary direct images. These assertions require no constructibility hypothesis.

We will use the following consequence of F.1. If \(M\) is c-soft and \(Z\subset X\) is closed, then every compactly supported section of \(M|_Z\) extends to a compactly supported section of \(M\) on \(X\). Choose a compact neighborhood \(T\) of its support. On \((Z\cap T)\cup\partial T\), prescribe the given section and zero, respectively. They agree on the intersection because the support lies in \(\operatorname{int}T\). The finite closed gluing and compact extension of F.1 extend these data; cutting off across \(\partial T\) gives the required section, and it agrees with the original section on all of \(Z\), including the zero germs outside \(T\).

For \(y\in Y\), let \(X_y=f^{-1}(y)\). Restricting a section with proper support defines

This map is injective. If a representative \(s\) restricts to zero on \(X_y\), the closed image of its proper support misses \(y\); on the complementary neighborhood its restriction is zero. If \(F\) is c-soft, (H.2) is also onto: \(X_y\) is closed, so the compact extension just proved extends a compact fiber section to one on \(X\). A compact support is proper over \(Y\). For arbitrary \(F\), take the beginning \(0\to F\to I^0\to I^1\) of the Godement resolution. Both \(I^0,I^1\) are c-soft. Both sides of (H.2) are left exact in \(F\), since restriction to the fiber is exact. The two isomorphisms for \(I^0,I^1\) identify their kernels, proving that (H.2) is an isomorphism for every sheaf.

H.2. Derived fibers, bounds, and arbitrary proper-support base change

An injective sheaf on \(X\) restricts to a c-soft sheaf on the closed fiber, by F.1. Restricting an injective resolution to the fiber is exact, and that restricted resolution therefore computes compact-support cohomology. Apply (H.2) termwise to get the canonical derived fiber formula

The same proof works for a bounded-below complex, using the acyclic-resolution comparison proved in F.1. In particular, a sheaf which is c-soft on each fiber is \(f_!\)-acyclic. Bounded-below complexes of such sheaves compute \(Rf_!\) termwise.

If \(\dim_cX\leq N\), its closed fibers have dimension at most \(N\) by F.2. Thus \(R^qf_!F=0\) for \(q>N\) for every sheaf \(F\). Finitely many truncation triangles now show that

This is a bound for sheaves without a finiteness assumption on their stalks. It does not imply that the resulting cohomology sheaves are constructible.

Consider the Cartesian square with \(f':X'=X\times_Y Y'\to Y'\), \(g:Y'\to Y\), and \(g':X'\to X\). Pullback of a proper support is proper by H.1. Pulling back sections consequently gives the natural morphism

It may equivalently be formed by the inverse/direct-image adjunction units; on a local representative it is precisely restriction to the pullback support. At \(y'\), (H.2) identifies this map with the identity on compact sections of the same fiber \(X_{g(y')}=X'_{y'}\), with its same restricted sheaf. It is therefore an isomorphism.

To derive it, choose an injective resolution \(I\) on \(X\). The restriction of \(g'^*I\) to every \(f'\)-fiber is the restriction of \(I\) to the corresponding \(f\)-fiber and is c-soft. Thus \(g'^*I\) is termwise \(f'_!\)-acyclic, while \(g^*\) is exact. Applying (H.5) termwise proves

This proves the actual base-change map on bounded-below complexes. Pasting two squares gives the same map: both underived maps pull back the same local section to its iterated pullback support, and the acyclic comparisons used to derive them are natural. If \(f\) is proper, \(f_!=f_*\), so this includes classical proper base change for ordinary direct image.

H.3. Composition and c-soft images

Proper direct image preserves c-soft sheaves. Let \(M\) be c-soft on \(X\), and take a section of \(f_!M\) over a compact \(K\subset Y\). By F.1 it is represented on an open neighborhood \(V\) of \(K\), hence by a section \(s\) of \(M\) over \(f^{-1}V\) with support proper over \(V\). Choose a compact neighborhood \(T\) of \(K\) inside \(V\). The restriction to the closed set \(f^{-1}T\) has compact support. The compact closed-section extension in H.1 extends it to a compactly supported section \(t\) on \(X\). As a section of \(f_!M\), it agrees with the original section on \(\operatorname{int}T\), and thus extends its germs on \(K\). This proves c-softness of \(f_!M\).

For composable \(X\xrightarrow fY\xrightarrow gZ\), the ordinary identity \(g_*f_*=(gf)_*\) restricts to

Here is the support check in both directions. A section of \(F\) over \((gf)^{-1}V\) has a closed support \(S\). If \(S\to V\) is proper, then \(S\to g^{-1}V\) via \(f\) is proper: the inverse image of a compact \(T\subset g^{-1}V\) is a closed subset of the compact inverse image of \(g(T)\). Thus the section defines one of \(f_!F\). Its support is \(f(S)\), by the fiber isomorphism (H.2); this image is closed. Its inverse image over a compact subset of \(V\) is the image of the corresponding compact subset of \(S\), so it is proper over \(V\). Conversely, if \(S\to g^{-1}V\) is proper and \(f(S)\to V\) is proper, the inverse image in \(S\) of a compact subset of \(V\) is obtained by two successive compact inverse images. This proves (H.7).

An injective resolution on \(X\) has c-soft terms. Its image under \(f_!\) has c-soft terms by the first paragraph, hence is termwise \(g_!\)-acyclic by H.2. The bounded-below acyclic comparison therefore derives (H.7) to

Threefold composition and the identity map are compatible because the underived identifications preserve the same section and support, and the resolution comparisons are natural. This also proves \(R\Gamma_c(Y,Rf_!F)=R\Gamma_c(X,F)\).

H.4. Sums and the projection formula

Proper direct image commutes with filtered colimits. Indeed, after taking a stalk, this is (H.2) and the compact-section colimit theorem F.4 on the fiber; inverse image commutes with these colimits because it is computed on stalks. In particular \(f_!\) commutes with arbitrary direct sums.

Multiplying a section with proper support by a pulled-back section defines

Its support is contained in the original proper support. On a fiber over \(y\), the pulled-back sheaf \(G\) is the constant sheaf with value \(G_y\). The stalk of (H.9) is therefore \(\Gamma_c(X_y,F|_{X_y})\otimes G_y\to \Gamma_c(X_y,F|_{X_y}\otimes\underline{G_y})\). This is an isomorphism: choose a basis of \(G_y\), express the tensor sheaf as a sum of copies of \(F|_{X_y}\), and use the compact-section direct-sum statement in F.2. The resulting map is the original multiplication map, independent of the chosen basis. Hence (H.9) is an isomorphism of sheaves.

For complexes, replace \(F\) by a bounded-below injective resolution \(I\), and use a bounded-below representative of \(G\). Each term of \(I\otimes f^*G\), restricted to a fiber, is a finite sum of tensor products of a c-soft sheaf with a constant vector space. Such a tensor product is a direct sum of c-soft sheaves, hence c-soft. Thus this complex computes its proper-support image termwise. Over a field tensoring preserves quasi-isomorphisms: stalkwise an acyclic complex of vector spaces contracts, and its tensor product contracts by the same homotopy with the Koszul sign convention. Applying (H.9) termwise gives

This proves the projection formula for the stated bounded-below complexes, with arbitrary coefficient sheaves, rather than only for finite-rank local systems.

H.5. The general classical exceptional inverse image

Let \(M\) be c-soft on \(X\). F.3 proves that \(F\otimes M\) is c-soft for every sheaf \(F\). Thus \(T_{M,f}(F)=f_!(F\otimes M)\) is exact: apply the long exact sequence of \(Rf_!\) and the acyclicity of its kernel tensor term. It preserves direct sums by H.4, hence all colimits by the cokernel presentation of a colimit used in F.3.

Its right adjoint on a sheaf \(G\) on \(Y\) is the sheaf

The proof is the actual section presentation of F.3 with \(T_{M,f}\) in place of \(T_M\). For clarity, apply this exact, sum-preserving functor to the open-cover presentation of \(\Lambda_U\); taking Hom into \(G\) gives the equalizer which proves the sheaf axiom for (H.11). A presentation of arbitrary \(F\) by sums of \(\Lambda_U\), followed by such a presentation of its kernel, identifies both kernels and gives

If \(G\) is injective, exactness of \(T_{M,f}\) implies injectivity of \(f^{\#}(M,G)\).

Choose the bounded c-soft resolution \(\Lambda_X\to L\) of F.4 in degrees \(0,\ldots,N\). Replace a bounded-below \(G\) on \(Y\) by an injective resolution \(J\). Define

This is a bounded-below complex of injectives. The mixed terms in its square cancel exactly as in F.12. The augmentation \(F\to F\otimes L\) is a quasi-isomorphism, and its total terms are c-soft, so \(f_!(F\otimes L)\) computes \(Rf_!F\). Currying (H.12) identifies the two mapping complexes: on \(F^i\otimes L^p\), a degree-\(n\) map has differential \(d_Jh_{i,p}-(-1)^nh_{i+1,p}d_F-(-1)^{n+i}h_{i,p+1}d_L\) on both sides. Hence

As in F.4 the mapping complexes may be unbounded; the adjunction objects themselves lie in \(D^+\). Quasi-isomorphisms and different choices of resolution give the same right adjoint by this Hom characterization. Its counit \(Rf_!f^!G\to G\) is the trace associated to the identity of \(f^!G\). Equation (H.8) identifies the right adjoint of a composite with the composite of these right adjoints, and identifies its counit with the composed counit. In particular \(\omega_X=f^!\omega_Y\). For an open immersion this agrees with \(j^!=j^*\), and for a closed immersion with B.2's \(i^!\), by uniqueness of the respective right adjoints.

H.6. Duality identities before constructible biduality

For bounded \(A\) and bounded-below \(B\) on \(Y\), the constructed operations satisfy

Map a bounded-below \(H\) on \(X\) into either side. The adjunction (H.14), tensor–Hom currying, and (H.10) give in order \(R\operatorname{Hom}_Y(Rf_!H\otimes A,B)\), \(R\operatorname{Hom}_Y(Rf_!(H\otimes f^*A),B)\), and \(R\operatorname{Hom}_X(H\otimes f^*A,f^!B)\). These are precisely the mapping complexes into the right side. All targets are bounded below because \(A\) is bounded; the injective Hom construction of F.5 justifies the internal derived Hom. Naturality and Yoneda prove (H.15), with the map obtained from evaluation and the adjunction counit.

For bounded \(K\) on \(X\) and bounded-below \(B\) on \(Y\), the same argument, now testing with \(H\) on \(Y\), gives

Indeed, its mapping complexes are successively \(R\operatorname{Hom}_X(f^*H\otimes K,f^!B)\), \(R\operatorname{Hom}_Y(Rf_!(f^*H\otimes K),B)\), and \(R\operatorname{Hom}_Y(H\otimes Rf_!K,B)\). Bound (H.4) makes \(Rf_!K\) bounded, so this internal-Hom target again lies in \(D^+\). Taking the structural dualizing complexes in (H.15)–(H.16) proves

Neither identity uses constructible biduality. The model (F.12) for \(\omega_X=a_X^!\Lambda\) is a bounded complex of injectives: the coefficient complex has its sole term in degree zero and \(L\) has only finitely many degrees. Consequently \(D_XK\) is bounded for bounded \(K\), by the internal-Hom construction of F.5. Equation (H.4) then also bounds \(Rf_!D_XK\). If evaluation is additionally known to be an isomorphism for both \(K\) and \(Rf_!D_XK\), applying the second identity to \(D_XK\) and then dualizing gives \(D_YRf_*K\simeq Rf_!D_XK\). This proves the exact reduction for the reverse exchange. Appendix J proves the biduality hypothesis for every complex constructible on a given finite Whitney stratification. K.6–K.8 prove the required algebraic refinements and proper-support image constructibility, then establish the reversal for every classical algebraic image at the stated scope. Likewise (H.13) constructs \(f^!\) on \(D^+\), and K.8 proves its preservation of bounded algebraic constructibility using the pullback exchange and canonical biduality. No such preservation is inferred merely from the existence of the adjunction.

For proper-support operations, see Mathew, §§2.3–2.6 and 4, and Schnürer–Soergel, §§3–7.

Appendix I. Control tubes and normal local triviality

This appendix proves the geometric local-triviality input for a given finite Whitney stratification. Work locally in a smooth real manifold, with smooth meaning \(C^\infty\), and let \(Z\) be a closed subset partitioned into finitely many smooth locally closed strata. The frontier of a stratum is a union of smaller strata. We write \(S<T\) when \(S\subset\overline T\setminus T\). The local assertions also apply to a locally finite stratification by restricting to a neighborhood meeting finitely many strata. Existence of a finite algebraic Whitney stratification and finite cohomology for constructible coefficients on its links are separate assertions; they are not consequences of the definition. Appendix J proves the latter coefficient assertion after the geometry constructed here; K.6 proves the algebraic refinement assertion.

I.1. The elementary smooth tools used below

We record the smooth constructions needed in the argument. They do not require a triangulation theorem. A second-countable smooth manifold has a compact exhaustion \(K_n\subset\operatorname{int}K_{n+1}\): cover it by countably many relatively compact coordinate balls and successively add finitely many such balls covering the preceding compact set. On each compact annulus \(K_n\setminus\operatorname{int}K_{n-1}\), choose finitely many smaller coordinate balls whose closures lie in prescribed covering sets and in \(\operatorname{int}K_{n+1}\setminus K_{n-2}\). The resulting family is locally finite. This proves the refinement and closed-shrinking properties used here.

The function \(b(t)=e^{-1/t}\) for \(t>0\), and zero for \(t\leq0\), is smooth. Its successive right derivatives are polynomials in \(1/t\) times \(e^{-1/t}\), and tend to zero: use \(e^s\geq s^m/m!\) with \(m\) larger than the polynomial degree. Translates and products give smooth bumps supported in the preceding coordinate balls and positive on the smaller balls. Divide them by their locally finite positive sum to get a partition of unity. In particular every positive continuous function has a positive smooth minorant: choose each bump's constant coefficient below the function on that bump's support, then sum using the partition. The same construction supplies a smooth metric and a proper smooth exhaustion function. For the latter, take smooth cutoffs \(\chi_n=1\) near \(K_n\), supported in \(\operatorname{int}K_{n+1}\), and sum \(1-\chi_n\). Near any point all but finitely many summands vanish; outside \(K_{m+1}\) the sum is at least \(m\).

The smooth metric also supplies a compatible distance by taking infima of lengths of piecewise smooth paths on each connected component. On a compact coordinate ball its quadratic form is bounded above and below by positive multiples of the Euclidean form. Straight segments give the upper local distance bound; every path exiting a larger coordinate ball must cross a shell of positive Euclidean width, giving the lower local bound. Thus this distance induces the manifold topology. Distinct components can be assigned a bounded positive separation after bounding the distance within each component.

The smooth inverse-function argument can be obtained directly by contraction. After translating and applying an invertible linear map, write a map as \(F(x)=x+R(x)\), with \(\|dR\|<1/2\) on a closed ball. For small \(y\), iteration of \(x\mapsto y-R(x)\) stays in that ball and converges, since successive differences decrease geometrically. It gives the unique inverse. Subtracting the equations for two values gives continuity and a Lipschitz bound; taking difference quotients gives derivative \((dF)^{-1}\). Repeated differentiation proves smoothness. The same estimates, uniform on a compact parameter set, give the parameter version. If \(dq\) is onto, complete its rows to an invertible coordinate differential and apply this result: \(q\) becomes a coordinate projection. If \(S\) is a submanifold and \(q|_S\) is onto, first choose coordinates on \(S\) with these first coordinates, extend them to a submanifold chart, and use \(q\) itself for the first coordinates. Thus simultaneously

We also need the elementary flow theorem. In a compact coordinate ball a smooth vector field \(V\) has bounds \(|V|\leq M\), \(\|dV\|\leq L\). For \(aM\) smaller than the available radius and \(aL<1\), the integral operator \(x(t)\mapsto x_0+\int_0^tV(x(s))\,ds\) is a contraction on the closed set of continuous paths staying in that ball, with the supremum norm. Its iterates converge geometrically to a differentiable solution; the integral equation gives its derivative. The same argument proves uniqueness and continuous dependence. More explicitly, iterating the inequality for two solutions bounds their difference by \(e^{L|t|}|x_0-y_0|\); the remainder after \(m\) iterations is bounded by a constant times \((L|t|)^m/m!\), which tends to zero. The differentiated integral equation for the initial-value derivative is a linear integral equation with the same contraction estimate. Higher differentiated equations have already constructed lower derivatives in their inhomogeneous terms, so induction proves smooth dependence of every order. Uniqueness gives the composition law and the inverse flow at negative time. Patching the coordinate solutions gives a maximal open time interval on a manifold. At a finite endpoint its trajectory leaves every compact subset of that manifold: otherwise a convergent subsequence in a compact set enters a coordinate ball with a uniform positive existence time and extends the trajectory. These are all the smooth-flow facts used below.

I.2. Whitney regularity and the tubular submersion

In Euclidean coordinates Whitney (b) for \(T,S\) says: if \(x_n\in T\), \(y_n\in S\) converge to \(p\in S\), their tangent planes converge to \(E\), and their secant lines converge to \(\ell\), then \(\ell\subset E\). It is invariant under smooth coordinate changes. Indeed, for a diffeomorphism \(h\), integration of its derivative on the short segment between the two points gives

The limiting tangent plane becomes \(dh_pE\), and the limiting secant becomes \(dh_p\ell\), since \(dh_p\) is invertible. Applying the inverse diffeomorphism proves both directions.

Condition (b) implies (a), namely \(T_pS\subset E\). For \(0\ne v\in T_pS\), choose a smooth curve \(c(t)=p+tv+o(t)\) in \(S\). With \(t_n=\sqrt{|x_n-p|}\), the secants from \(x_n\) to \(c(t_n)\), divided by \(t_n\), tend to \(-v\). Condition (b) puts \(v\) in \(E\). In coordinates flattening \(S\), the unit normal direction from \(x_n\) to its coordinate projection onto \(S\) also has a convergent subsequence. Condition (b) puts its limit in \(E\); it is nonzero and perpendicular to \(T_pS\). Consequently \(\dim T>\dim S\) for every incident distinct pair. The compactness used here is elementary: orthogonal projectors of fixed rank form a closed bounded subset of the finite-dimensional matrix space.

A tube of \(S\) is a diffeomorphism \(\phi\) from a neighborhood of the zero section in an inner-product vector bundle \(E\to S\) onto a neighborhood of \(S\), restricting to the identity on that section. Write

In an orthonormal bundle chart these are coordinate projection and squared normal distance. Such a chart exists by applying Gram–Schmidt to a local frame; its divisions are by positive smooth norms. For a Whitney pair \(T,S\),

Here is the rank argument. Use the tube chart and put \(r=|v|>0\), \(u=v/r\). Rank failure of \((\pi_S,r)|_T\) gives \(a_n\in T_pS\cong\mathbb R^{\dim S}\), \(b_n\in\mathbb R\), with \(|a_n|^2+b_n^2=1\), such that \(a_n+b_nu_n\) annihilates \(T_{x_n}T\). If failing points approach \(p\in S\), pass to limits of these vectors, the unit normals, and the tangent planes. Conditions (a) and (b) put both the whole horizontal space and \(u\) in the limiting plane. Thus \(a+bu\) belongs to that plane and its orthogonal complement, although its norm is one. This contradiction proves (I.4); replacing \(r\) by \(r^2\) does not change rank away from \(S\). A positive smooth minorant of the pointwise permitted radii makes the shrinking global. There are only finitely many relevant pairs, so it can be done for all of them at once.

I.3. Relative construction and gluing of tubes

We prove the precise relative tube fact needed for compatible controls. Suppose \(q:M\to P\) is smooth and \(q|_S\) is a submersion. A tube is called \(q\)-compatible if \(q\pi_S=q\). If such a tube is prescribed on an open \(O\subset S\), and \(C\subset O\) is closed in \(S\), it extends to a \(q\)-compatible tube on all of \(S\), preserving the prescribed tube on some neighborhood of \(C\). We prove preservation of the full embedding germ: two tubes agree there when an inner-product bundle isomorphism \(\psi_0:E_0\to E_1\), over the identity on the base, satisfies \(\phi_1\psi_0=\phi_0\) near the zero section. This implies equality of \(\pi_S,\rho_S\) after shrinking.

The converse implication would be false. On \(\mathbb R^2\), let \(R_\theta\) be rotation through \(\theta\), and compare the tube embeddings of the point \(\phi_0(v)=v\) and \(\phi_1(v)=R_{|v|^2}v\). The latter is a smooth diffeomorphism, with inverse \(w\mapsto R_{-|w|^2}w\); it preserves the norm and has derivative the identity at zero. Both tubes therefore have the same projection and radius functions. If a linear isometry \(\psi_0\) identified their embedding germs, differentiation would force \(\psi_0=1\), whereas \(\phi_1(v)\ne v\) for sufficiently small nonzero \(v\). We retain the full tube when protecting a prescribed region, so this ambiguity does not enter the construction.

Local existence follows from (I.1), taking the trivial normal bundle and the product coordinates. We first show that two local \(q\)-compatible tubes can be made to agree near a compact subset of a common chart by a diffeomorphism supported in that chart, fixing \(S\) and preserving \(q\). Let their embeddings be \(\phi_0,\phi_1\), and let \(\alpha_i:E_i\to TM|_S/TS\) be their normal differentials. Put

Then \(\psi\) is a bundle isometry and \(\beta^{-1}\psi=A^{-1/2}\) is positive self-adjoint. For completeness, a real symmetric operator has an orthonormal eigenbasis: maximize its quadratic form on the unit sphere, differentiate along that sphere to obtain an eigenvector, and induct on its invariant orthogonal complement. For a positive operator all eigenvalues are positive; this defines its unique positive square root. Smooth dependence of its inverse square root follows locally from the convergent binomial series for \(\alpha^{-1/2}(1-(1-A/\alpha))^{-1/2}\), choosing \(\alpha\) so that \(\|1-A/\alpha\|<1\). On a smaller neighborhood the series and each differentiated series converge uniformly. Thus (I.5) is smooth.

Use the simultaneous affine coordinates (I.1). On a small common domain define

Each map fixes \(S\) and preserves \(q\). Its normal differential is \(\alpha_0((1-t)1+tA^{-1/2})\), which is invertible. The parameter inverse-function argument gives a common neighborhood of the compact subset where every \(h_t\) is a diffeomorphism onto its image. Injectivity can be made uniform there: otherwise pairs of distinct points with equal images, approaching the compact zero section, have convergent subsequences and contradict the parameter local inverse at their common limit. Notice that \(h_0=1\), while \(h_1\phi_0=\phi_1\psi\). On a region where a full germ identification \(\phi_1\psi_0=\phi_0\) is already given, taking normal derivatives gives \(\alpha_1\psi_0=\alpha_0\). Thus \(\beta=\psi_0\), \(A=1\), and \(\psi=\psi_0\) there. Formula (I.6) is consequently constant in \(t\) on that region: \(\Phi_t=\phi_0\), so \(h_t=1\). Equality of projection and radius alone is not used to infer this identity.

To extend the germ to an ambient diffeomorphism, choose a smooth cutoff \(\chi\), equal to one near the compact set and supported in a small common image neighborhood. For \(s,t\) close define, in the affine chart and by the identity outside it,

These preserve \(q\), fix \(S\), and equal the identity wherever the full embedding germs agree through the given bundle isometry. Uniformly on the compact support, \(G_{s,t}-1\) has derivative norm less than \(1/2\) when \(|s-t|\) is small. Extending the difference by zero to Euclidean space, the contraction equation \(z=w-(G_{s,t}(z)-z)\) proves global invertibility; the inverse is smooth by I.1. Choose an integer \(m\) with \(1/m\) below that bound and compose \(G_{(m-1)t/m,t}\cdots G_{0,t/m}\). Near the compact set all cutoff factors are one, so this telescopes to \(h_t\). At \(t=1\) it gives the required ambient diffeomorphism. The support can avoid any closed protected subset disjoint from that compact set. This proves the local gluing assertion, including preservation of a previously agreed germ.

Here are the global extension details. Replace \(M\) by \(M\setminus(\overline S\setminus S)\), so \(S\) is closed. Choose a locally finite relatively compact chart cover as in I.1, with compact smaller pieces whose interiors cover \(S\). Starting from the prescribed tube, attach one chart at a time. At a finite stage let \(D\subset S\) be the closed protected set, consisting of \(C\) and the preceding protected compact neighborhoods, and let \(B\subset S\) be an open neighborhood on which the old tube is defined. Choose a smooth function \(0\leq\theta\leq1\), equal to one near \(D\) and zero outside \(B\). Such a function follows from I.1's locally finite bumps and closed shrinking: choose \(D\subset B_0\subset\overline B_0\subset B\), take a partition subordinate to \(B\) and \(S\setminus\overline B_0\), and sum the terms supported in \(B\).

Let the next compact piece \(K\) lie in a relatively compact open base region \(W_0\), with \(\overline W_0\) inside one simultaneous \(q\)-chart \(W\). Retain the old tube over \(B_1=\{\theta>1/3\}\) and use the chart's product tube over \(A_1=W_0\cap\{\theta<2/3\}\). Their union contains \(D\cup K\). The closure of their overlap is contained in the compact set

which avoids a neighborhood of \(D\). Apply (I.5)–(I.7) near \(Q\), with compact ambient support in \(W\) disjoint from \(D\), to send the old tube embedding to the product tube embedding through \(\psi\). Such support is possible because \(D\) is closed in \(M\) and \(Q\) is compact and disjoint from it. After shrinking the normal radii, the full embedding germs agree throughout the overlap \(A_1\cap B_1\). Glue their inner-product bundles by \(\psi\) there and glue their embeddings. There are only two pieces at this step; the transition and its inverse are the displayed bundle isometries. The new embedding remains \(q\)-compatible, and it is the prescribed old embedding near \(D\), where the ambient diffeomorphism is the identity. Choose an additional closed neighborhood of \(K\) inside \(A_1\cup B_1\) and protect it in all later stages.

Protect successively the earlier compact pieces and slightly larger closed neighborhoods of them. Their interiors exhaust \(S\). Hence every point has a neighborhood on which subsequent tube germs are unchanged. This produces a smooth bundle and embedding germ along the entire zero section, retaining the original germ near \(C\). A single positive radius makes that germ an embedding: take a proper exhaustion \(u\) of \(M\), shrink so that \(|u(\phi(v))-u(\operatorname{base}v)|<1\), and consider successive compact annuli of \(S\). Colliding images must have base exhaustion values differing by less than two. On each corresponding larger compact annulus, the compactness/inverse-function argument used above supplies a uniform injectivity radius. A positive smooth minorant below these locally finite bounds excludes every collision. Shrink once more to remain inside the domain of the germ. This establishes the claimed global relative tube, with no appeal to a tubular-neighborhood existence theorem.

I.4. Compatible control data

Suppose \(f:M\to P\) is smooth and its restriction to every stratum of \(Z\) is a submersion. We construct tubes for the given finite Whitney stratification satisfying

where the compositions are defined. The construction uses only I.1–I.3. We explain the shrinking choices as well as the induction. Two incomparable strata have disjoint closures after removing their frontiers; distance neighborhoods there separate them. Thus their tubes can be chosen disjoint. A tube of \(S\) can also avoid every other stratum not lying above \(S\): none of the closures of those finitely many strata meets \(S\). Radius minorants implement these separations. Further shrinkings retain equations already established.

We additionally shrink each tube of \(R\) so that \(d(m,\pi_Rm)<d(\pi_Rm,\partial R)/3\) when \(\partial R\ne\varnothing\). This follows from continuity along the zero section and a positive radius minorant on \(R\). The triangle inequality gives \(d(m,\pi_Rm)<d(m,\partial R)/2\). Consequently, as a point in that tube approaches \(s\in\partial R\), its projection also approaches \(s\). This ensures that shrinking a lower tube can make all the required higher projections lie in its original domain; it prevents an uncontrolled jump of a higher projection toward a distant base point. Finitely many pairs can be treated simultaneously.

Induct upward on stratum dimension. Assume the tubes of smaller strata have been constructed, with (I.4) for every larger stratum. For a new stratum \(T\), first construct its tube over the part of \(T\) meeting those smaller tubes. Proceed downwards through their dimensions. At one stage let \(O\subset T\) be the union of the already treated, larger-dimensional lower tubes intersected with \(T\), and suppose a tube on \(O\) preserves their projection and radius. Shrink its domain so that every point and its projection lie in at least one of those treated tubes. This is possible pointwise since they cover \(O\), followed by the radius-minorant construction.

Now consider a lower stratum \(S<T\) of the next dimension. On \(U_S\cap T\) the map \(q_S=(\pi_S,\rho_S)\) is a submersion by (I.4). On its overlap with the already constructed tube on \(O\), that tube is \(q_S\)-compatible after shrinking. Indeed choose a treated stratum \(R\) whose tube contains the point and its projection. When it also meets the \(S\)-tube, comparability and dimensions give \(S<R\). The existing equations give

All these compositions can be kept in their domains: near a base point each of the finitely many needed memberships is open; shrink the new tube there, then take a positive minorant. Choose a closed smaller region \(C_O\subset O\) retaining the intersections with sufficiently smaller treated tubes. Such a choice is possible because the complement of \(O\) in \(T\) cannot accumulate at a base point of a treated lower stratum: that stratum already has an ambient open tube. A compact-annulus shrinking in each treated tube gives closure inside \(O\), relative to \(T\).

Apply the relative extension of I.3 on \(U_S\cap T\), for \(q_S\), preserving the old tube near \(C_O\cap U_S\). Shrink the treated lower tubes to the retained smaller regions and glue. Distinct \(S\)'s at this stage have disjoint tubes, so these extensions do not interfere. This finishes the descending induction. There are finitely many strata and dimensions, so only finitely many such shrinkings occur.

The resulting tube near all lower strata is \(f\)-compatible: on a lower tube, \(f=f\pi_S\), and (I.9) gives \(f\pi_T=f\). Apply I.3 once more, now to \(f\), to extend it over all of \(T\), preserving a closed smaller neighborhood of the lower-tube intersections. Shrink those lower tubes accordingly. Finally apply I.4 to make the new tube's projection-radius map a submersion on every higher stratum. This completes the upward induction and proves (I.8).

For later compactness arguments, tubes may always be shrunk to closed-radius disk bundles with radius strictly below their original radius. Over a compact subset of \(S\), these disk bundles are compact: finitely many local bundle trivializations reduce this to closed bounded balls. Intersecting with the closed set \(Z\) retains compactness.

I.5. Controlled lifts and continuous stratified flows

A stratified vector field consists of a smooth field \(V_T\) on each stratum. It is controlled if, on a sufficiently small tube of every lower stratum \(S\),

For every smooth field \(W\) on \(P\), there is a controlled lift with \(df(V_T)=Wf\). Construct it upwards by dimension. On a new stratum \(T\), choose smaller closed tubes \(E_S\) around each \(S<T\), inside the neighborhoods where the already defined fields satisfy (I.10). Arrange additionally that \(\pi_R(E_S\cap E_R)\) lies in the original control neighborhood of \(S\) whenever \(S<R<T\). To justify this shrinking, first give \(E_S\) closure inside its original neighborhood relative to \(M\setminus\partial S\). At a point \(r\in R\) outside that original neighborhood there is then a neighborhood disjoint from \(E_S\); shrink the \(R\)-tube there. Inside it continuity of \(\pi_R\) gives the required membership. Repeat for the finitely many pairs. Incomparable closed tubes are disjoint as before.

At \(x\in T\), the indices with \(x\in E_S\) form a chain. If nonempty, take its largest member \(R\). Solve \((d\pi_R,d\rho_R)V_T=(V_R\pi_R,0)\) locally, using the submersion (I.4). A smooth right inverse is explicit: for a surjective matrix \(B\), use \(B^*(BB^*)^{-1}\). Equations (I.8), together with the already proved control of \(V_R\), imply every required smaller-stratum equation and \(df(V_T)=Wf\). Indices inactive at \(x\) can be excluded from a neighborhood of \(x\), since \(E_S\cap T\) is closed. If the chain is empty, solve just \(df(V_T)=Wf\). These constructions give local fields satisfying all constraints at every point where they are required. A partition of unity on \(T\) glues them: the constraints are affine linear with the same right-hand side, and the weights sum to one. This completes the lift.

The stratumwise smooth flows supplied by I.1 combine into a continuous local flow on \(Z\). We prove the two points that do not follow from the separate smooth flows. First, a trajectory cannot have a finite maximal endpoint while visiting a compact subset of \(Z\). If it did, choose a sequence of times tending to that endpoint with images tending to \(y\in S\). If the trajectory's stratum is \(S\), the ordinary coordinate-flow argument extends it. Otherwise the frontier condition makes \(S\) a lower stratum. Choose a compact coordinate neighborhood in \(S\) on which the lower flow exists for a fixed time \(\delta>0\), and a small closed tube over its compact flow image lying inside the control neighborhood. For a sufficiently late time \(t_i\), the trajectory is in this tube, with \(\rho_S=c_i>0\), and the remaining time is less than \(\delta\). For as long as it is defined, (I.10) forces its projection to be the lower trajectory and its radius to remain \(c_i\). It cannot leave the control neighborhood: these two conditions confine it to a compact subset of that neighborhood, so a first exit would contradict continuity. All later points of the convergent subsequence therefore have radius \(c_i\), contradicting their limit \(y\), whose radius is zero. This proves escape from compact sets at every finite maximal endpoint.

Second, fix a stratumwise flow value \(\Phi(t,x)\), with \(x\in S\). Choose a compact neighborhood \(K\subset S\) of \(x\) and a time margin so the lower flow of \(K\) over the enlarged time interval is compact. Choose a sufficiently small closed disk tube over that compact image, inside the control neighborhood. For \(z\) near \(x\), the same equations give

They hold first for short times, and then throughout the enlarged interval: a first exit is impossible and a finite maximal endpoint would contradict the compact-escape result just proved. Thus the domain of the combined flow is open near \((t,x)\). As \((s,z)\to(t,x)\), the right side of the second identity tends to \(\Phi_S(t,x)\), while the first tends to zero. Compactness of the disk tube implies that every convergent subsequence of the images has that same limit on its zero section; hence the images converge to it. This proves joint continuity. The group and inverse identities hold on each stratum, so the combined local maps are homeomorphisms preserving every stratum. No continuity of the ambient vector-field values across strata was assumed.

I.6. Proper stratified submersions and normal cones

If \(f|_Z:Z\to P\) is proper and a submersion on each stratum, then it is locally a product, by homeomorphisms preserving all strata. Choose a coordinate box in \(P\), centered at \(p\), and use I.4–I.5 to lift its coordinate fields. A trajectory lifting a coordinate segment in a compact smaller box stays in its compact inverse image. The compact-escape result proves existence for the entire segment. Compose these flows in the fixed order \(1,\ldots,d\) to define

Its inverse uses the negative flows in reverse order, together with the value of \(f\). Both are continuous by I.5 and preserve strata. The lifted fields need not commute; the specified inverse order is what proves the two composite maps are the identity. The same argument works over a whole coordinate box whenever the map is proper over that box, since each particular coordinate path is compact. This is the first isotopy lemma at the finite Whitney scope used here.

Now fix \(p\in S\subset Z\). Work in a neighborhood avoiding all strata not above \(S\), so \(S\) is closed there. Take its tube, a relatively compact coordinate box \(B\subset S\), and a constant \(\epsilon>0\) below the squared tube radius over \(\overline B\), small enough for (I.4) on every upper stratum. On the punctured tube,

is proper: the inverse image of a compact subset is closed in a compact disk bundle and has radius bounded away from zero and the outer boundary. It is a submersion on every stratum. Apply the proved isotopy lemma, using controls compatible with this \(q\), to obtain a product with the compact fiber \(L=q^{-1}(p,\epsilon/2)\). Adjoin the zero section and collapse \(L\times\{0\}\) to a vertex. The resulting map is a stratum-preserving homeomorphism

Here the cone parameter is squared normal distance. Continuity at the added vertex follows from compact disk-bundle compactness: if the base tends to \(s\) and the radius tends to zero, every cluster point is the unique zero-section point \(s\). The inverse is continuous there because a neighborhood of the vertex in a cone on a compact \(L\) contains all sufficiently small radii. The empty-link case gives just \(B\). Restricting \(B\) and the radius gives arbitrarily small such neighborhoods, since the tube is an actual ambient neighborhood and its compact fibers shrink to the zero section. This proves normal local triviality, including compatible shrinking in the displayed product.

The link inherits Whitney regularity. More generally, if a smooth map \(q\) is a submersion on every stratum, its level intersections have tangent spaces \(T_xT\cap\ker dq_x\). In a convergent pair of tangent planes the limit contains the tangent space of the lower stratum by (a), on which \(dq\) is onto. The kernels therefore have constant codimension and converge to the intersection of the limiting plane with \(\ker dq_p\); this follows by solving the surjective rows with \(B^*(BB^*)^{-1}\). Secants between points of the same level have limit in \(\ker dq_p\), by the derivative estimate (I.2) applied to \(q\). Original condition (b) then gives condition (b) for the intersections. For (I.13), the product identifies every nonempty upper piece with its whole link piece times the base box, so its original full frontier incidences restrict to full incidences on \(L\). The dimension of each link piece is \(\dim T-\dim S-1\), strictly below the dimension of \(T\).

These arguments supply the normal local-triviality theorem used in Sections 2 and 4 once a finite Whitney stratification is given. Appendix J next proves finite-dimensional link cohomology with constructible coefficients and the associated duality theorem. K.6–K.8 subsequently prove existence of algebraic Whitney refinements and preservation of algebraic constructibility by the classical operations. For control tubes, flows and local triviality, see Mather's Notes on Topological Stability, §§1–2 and 6–10; the first isotopy lemma is in §11.1.

Appendix J. Constructible coefficients on Whitney neighborhoods

Let \(X\) be a locally closed subset of a finite-dimensional smooth real manifold, with a given finite Whitney stratification \(\mathcal S\). We retain the Hausdorff, metrizable and finite-dimensional hypotheses of Appendices F–I. A complex is \(\mathcal S\)-constructible here if it is bounded and its cohomology sheaves restrict to finite-rank local systems on every stratum. The coefficient ring is a field \(\Lambda\).

This appendix proves the coefficient comparison, finite cohomology on compact links, biduality and preservation of this fixed constructibility system. The order of proof matters:

Thus finite link cohomology is proved after local evaluation, using G.1; it is not inserted as an assumption into that criterion. No existence theorem for algebraic Whitney refinements is used.

J.1. The coefficient comparison along an interval

Put \(I=[0,1]\). If \(E\) is a bounded complex on \(Y\times I\) and the cohomology sheaves of its restriction to each fiber \(\{y\}\times I\) are locally constant, then, for the projection \(p:Y\times I\to Y\), the counit is an isomorphism:

Here \(Y\) has the classical hypotheses above; no local constancy in the \(Y\) direction is assumed. The map \(p\) is proper: its inverse image of a compact set is a product of two compact spaces. Proper base change (H.6) identifies the counit's stalk at \((y,t)\) with evaluation

A local system on \(I\) is constant. Indeed Appendix A.4's transport identifies its fibers by the unique endpoint-fixed homotopy class of paths in \(I\); linear interpolation contracts any two such paths relative to their endpoints. The constant-sheaf singular resolution in A.2 and C.1–C.2 applies also to an interval with its endpoints: their neighborhood basis consists of intervals or half-intervals, with the same straight-line contraction. Its prism identity gives \(H^0(I,\underline V)=V\) and \(H^q(I,\underline V)=0\) for \(q>0\), and evaluation is the displayed isomorphism in degree zero. Apply the finitely many truncation triangles to the fiber complex. They show that (J.2) is an isomorphism in every degree. This proves (J.1). They also show that \(Rp_*E\) has the same cohomological bounds as \(E\), so every complex just used remains bounded.

There is a useful map-level consequence. Suppose \(H:Y\times I\to X\) is a continuous homotopy such that, for each \(y\), its track \(H(y,I)\) lies within one stratum of \(X\). For constructible \(F\), set \(E=H^*F\), and write \(h_t(y)=H(y,t)\). Exact inverse image identifies the fiber cohomology sheaves of \(E\) with the pulled-back local systems on that track. Restricting (J.1) to the two endpoints gives natural isomorphisms

These also compare the maps on derived ordinary sections. The adjunction map \(R\Gamma(X,F)\to R\Gamma(Y\times I,H^*F) =R\Gamma(Y,Rp_*H^*F)\), followed by either endpoint map, is the original pullback by \(h_0\) or \(h_1\). This follows directly from composition of the inverse/direct-image adjunction units in H.1. For a constant homotopy the comparison is the identity. Restricting \(Y\), or restricting the parameter to a subinterval, restricts the same evaluation maps. Thus this construction controls the actual restriction morphisms, not just their cohomology dimensions.

In particular, suppose \(V\subset U\) and a map \(r:U\to V\) is joined to the identity of \(U\) by such a homotopy, all of whose maps also carry \(V\) into \(V\). Then

is an isomorphism. To exhibit its inverse, pull back sections along \(r\), then use (J.3) to identify \(r^*(F|_V)\) with \(F\). The composite on \(U\) is the identity by the preceding compatibility with pullback. The composite on \(V\) is the identity by the restriction of the same homotopy to \(V\). This supplies both inverse identities.

We will also use a product form of this argument. Let \(B\) be a convex open box or an interval, fix \(b_0\in B\), and suppose \(E\) on \(Y\times B\) is constructible for strata \(T\times B\). The homotopy \((y,b,t)\mapsto(y,(1-t)b+tb_0)\) has its tracks in those strata. Formula (J.3) gives

For the assertion on sections, projection and the slice are inverse up to this very homotopy; use its compatibility with their pullback maps as in (J.4). The same proof applies to \(\operatorname{pr}_Y^*G\) for any bounded \(G\), because its fiber restrictions are constant. Comparisons for different slices are the interval evaluation maps. In particular restrictions between subintervals, after transport to one fixed slice, are identities. This last assertion also follows by using a larger interval containing both slices and factoring all maps through its section complex.

J.2. Ordinary local cohomology and canonical biduality

At \(x\in S\), I.6 supplies one fixed stratum-preserving chart \(B\times C_R(L)\), taking \(x\) to \((0,v)\), with \(L\) compact. Here \(C_R(L)=(L\times[0,R))/(L\times\{0\})\), and \(v\) is its vertex; the cone coordinate is the squared normal radius of I.6. Smaller boxes and smaller positive radii give a cofinal basis of neighborhoods of \(x\).

For \(0<r<R\), take homothetic boxes and cones \(U_r=B_r\times C_r(L)\subset U_R=B_R\times C_R(L)\). The map

maps \(U_R\) into \(U_r\). Interpolate \(\lambda\) linearly from \(1\) to \(r/R\). Every track stays in the same stratum: a positive radial coordinate remains positive, its link point is fixed, and the vertex stays in the vertex stratum. The homotopy preserves \(U_r\) as well. Thus (J.4) proves that restriction between these neighborhood section complexes is an isomorphism. The case of an empty link is the same box contraction.

Exact filtered colimits on an injective resolution give \(\varinjlim_r H^q(R\Gamma(U_r,F))=H^q(F_x)\), as in G.1. Since all the restriction maps are isomorphisms, each original evaluation map is an isomorphism:

The stalk complex is perfect, by boundedness and finite-dimensional cohomology. G.1 therefore applies at every point and proves the canonical map

We have used neither finite link cohomology nor constructibility of \(D_XF\) here. Its boundedness, needed in G.1, comes from F.4–F.5: \(\omega_X\) has a bounded injective model, and internal Hom from a bounded complex into that model is again bounded. The same applies to the second dual. These are results for arbitrary coefficient sheaves over the field.

J.3. Compact constructible cohomology is finite

Suppose now that \(X\) is compact. The structural duality of F.14–F.16, applied twice, gives

Ordinary and compact supports agree on \(X\). Under (J.9), the sections of the map in (J.8) are the linear-complex evaluation map: both are obtained by currying evaluation followed by the structural counit. The sign is the degree sign checked in G.1. Thus \(R\Gamma(X,F)\) is canonically isomorphic to its algebraic double dual.

Here is the finite-dimensional conclusion, including the necessary linear algebra. For a vector space \(V\), its evaluation \(V\to V^{\vee\vee}\) is onto only if \(V\) is finite-dimensional. If \(V\) had an infinite basis \(\{e_i\}\), choose distinct \(e_{i_n}\), \(n\geq1\). In \(V^\vee\), let \(W\) be the span of all the coordinate functionals \(e_i^\vee\), and let \(a\) take value \(1\) on every \(e_{i_n}\) and \(0\) on the other basis vectors. Then \(a\notin W\). Define a functional on \(W+\Lambda a\) to be zero on \(W\) and one on \(a\), and extend it to \(V^\vee\) by extending a basis. This extension cannot be evaluation at any \(v\in V\): vanishing on all the coordinate functionals would force \(v=0\), while its value on \(a\) is one. This proves the assertion. The basis extension used here follows from the maximal-linearly-independent-set argument: a maximal independent extension must span, since otherwise one more vector could be added.

Linear dual is exact over a field by that same extension argument. Hence \(H^q(P^{\vee\vee})=H^q(P)^{\vee\vee}\), with the map induced by evaluation, for every complex \(P\). Apply this to \(P=R\Gamma(X,F)\) and (J.9). Its cohomology groups are finite-dimensional. They vanish outside a finite range by H.4, applied to the proper structural map. We have proved

In particular this holds on every link \(L\) from I.6: that link is compact, has the inherited finite Whitney stratification proved there, and is a closed subset of its smooth level manifold. No triangulation or unproved finite-type assertion about the individual open strata is required in this deduction.

J.4. The general normal attaching and support complexes

Consider the normal slice \(C_R(L)\) in I.6. Let \(F_0\) be a constructible complex on it. On the puncture, (J.5) gives \(j^*F_0=p_L^*J_0\), with \(J_0\) the restriction to one link slice. Set

Both \(A\) and \(B\) are perfect, the latter by (J.10). The map \(a\) is the vertex restriction of the actual unit \(F_0\to Rj_*j^*F_0\). The interval comparisons of J.1 identify \(i^*Rj_*j^*F_0=B\), compatibly with decreasing the outer radius.

The same localization proof as (G.7) now applies to this arbitrary stratified link:

To check it, use the two units and their natural commuting homotopy to map into the displayed cone fiber. On the puncture the map is the identity. At the vertex its target is \(\operatorname{Fib}(B\oplus A\to B)\); the identity \(B\)-pair contracts, leaving \(A\), and the induced map on \(A\) is the identity. This proves the quasi-isomorphism on every stalk.

Ordinary sections of the cone are \(A\), either by (J.7) or by canceling this same pair in (J.12). Sections of any annulus \(L\times(r,R)\) are \(B\), by (J.5); the restriction map from the cone is \(a\). For the compact closed subcone \(K_r\), support localization therefore gives

Changing \(r\) restricts the complementary annulus, which is the identity on \(B\) under the interval comparison. Thus the support-extension maps in (J.13) are identities in this model. The \(K_r\), \(r\uparrow R\), are cofinal among compact subsets of the open cone. Taking their union on an injective resolution proves

For the point costalk use the puncture rather than an annulus in the support fiber; the comparison from the puncture to each annulus is again the identity on \(B\). This identifies (J.14) with the actual support-extension morphism. Extension by zero between smaller and larger open cones also induces the identity on this model. In particular all the groups in (J.14) are finite-dimensional. Empty links give \(B=0\) and the same formulas.

J.5. Duality preserves the given Whitney stratification

Return to a fixed product neighborhood \(B_d\times C_R(L)\) of a point of a real \(d\)-dimensional stratum. Contracting only the box factor in (J.5) gives \(F=\operatorname{pr}_C^*F_0\) on this neighborhood. This is an isomorphism of complexes, so it includes the attaching data along all the link strata. Write \(C_0\) for (J.14).

For an oriented open box \(V\subset B_d\), let \(p:V\times C_r(L)\to C_r(L)\). Proper-support base change from \(V\to\{*\}\), followed by the projection formula H.10, gives

The first identification is C.5's compact-support calculation for the open box with its chosen orientation, transported by H.6. Composition H.8 and (J.14) now give

These comparisons have the required transition maps. For an inclusion of smaller oriented boxes, C.5 and F.5 identify extension by zero with preservation of the same local orientation generator. For smaller cone radii, it is the identity by (J.13)–(J.14). Base change, composition and the projection map are natural in these inclusions; hence the resulting maps in (J.16) are the identity under a fixed coordinate orientation.

The structural pairing (F.16) consequently yields

with identity restriction maps for these boxes and radii. For points on \(B_d\times\{v\}\), such neighborhoods form a basis, with boxes allowed to be centered at each point of the fixed larger box. Taking stalks and their restriction identifications shows that each cohomology sheaf of \(D_XF\) restricts to a constant finite-dimensional sheaf there, with the groups of \(C_0^\vee[d]\). This is a sheaf identification: the maps arise from the same compact-support pairings and orientation class, so their restrictions agree on smaller boxes. Changing a coordinate orientation inserts its usual sign and still gives a local system.

Every stratum is covered by these boxes. Boundedness was established before (J.8). We have therefore proved

with the canonical evaluation isomorphism. This proves the classical part of Proposition 2.4 for a given Whitney stratification.

F.20 applied to \(D_XF\), followed by (J.8), gives \(i_x^!F=(i_x^*D_XF)^\vee\). The latter stalk is perfect by (J.17). Dualizing and using its finite evaluation proves the other point exchange \(i_x^*D_XF=(i_x^!F)^\vee\). Together with F.20 this proves both identities in (2.1), including finiteness of the point costalks, at this classical scope.

J.6. Open and closed operations within the fixed stratification

Let \(j:U\hookrightarrow X\) be an open union of strata and \(K\in D^b_{\mathcal S|_U}(U)\). Extension by zero is constructible: it is \(K\) on the retained strata and zero on the others. The already proved identity \(D_Xj_!=Rj_*D_U\), applied to \(D_UK\), and (J.18) give

No independent boundary-image finiteness is assumed here. For a closed union of strata \(i:Z\hookrightarrow X\), exact \(i_*\) and \(i^*\) preserve constructibility. Apply F.20 to \(D_XF\) and use (J.18) on \(X\) to obtain

The inherited stratification on \(Z\) has the same Whitney and frontier properties, so J.18 applies there too. Locally closed unions are composites of such open and closed inclusions; the already proved composition identities give the corresponding assertions for them. Tensor products preserve constructibility on stalks and on stratum trivializations. Finally tensor–Hom currying against the bounded injective dualizing model and (J.18) identify

Both sides are bounded, and the right side is constructible by what was just proved. Thus internal Hom preserves the fixed constructibility system as well.

J.7. Proper images when compatible Whitney data are supplied

Here is the precise geometric image statement available without an algebraic refinement-existence theorem. Suppose \(f:X\to Y\) is proper, both spaces have given finite Whitney stratifications, and for every target stratum \(T\), the map \(f^{-1}T\to T\) has a smooth ambient extension and is a submersion on each of its source strata. The source strata in \(f^{-1}T\) form its inherited finite Whitney stratification. Then \(Rf_*F=Rf_!F\) is constructible on the given target stratification.

Indeed proper base change restricts this complex to the proper map over \(T\). By I.6 that map is locally a stratum-preserving product \(Z\times B\to B\), with \(Z\) a compact fiber. The kernel and secant argument of I.6 gives the induced Whitney stratification on \(Z\). Formula (J.5) identifies the restricted coefficient complex with the pullback of its fiber restriction \(F_Z\). Proper base change from \(Z\to\{*\}\) then identifies the image on \(B\) with the constant complex \(R\Gamma(Z,F_Z)\). It is perfect by (J.10); the uniform cohomological bound is H.4. This proves finite local constancy along every \(T\).

K.6–K.7 subsequently construct the compatible algebraic Whitney stratifications used in this statement. K.8 uses affine graph compactifications and finite open-cover descent to prove nonproper classical image constructibility without a global compactification theorem. The étale and normalized adic constructions have separate coefficient and geometric hypotheses.

For general homotopy invariance, see Haine–Porta–Teyssier, Theorem 0.4 and Section 5. At the bounded field-coefficient scope used here, J.1 gives a direct proof from proper base change and the interval calculation already established in this lesson. J.2–J.7 then give the local stabilization, compact finiteness, normal-support comparison and duality arguments themselves.

Appendix K. Algebraic Whitney refinements and classical images

In this appendix varieties are reduced separated schemes of finite type over \(\mathbb C\), considered also with their usual complex topology. Nilpotents do not change the classical sheaf spaces. We prove existence of finite algebraic Whitney refinements, compatible stratifications of algebraic maps, and the classical constructibility statements of Section 3. The proof is for field coefficients in its sheaf-theoretic part. It does not supply an étale comparison.

K.1. The elementary algebraic tools

We first give the algebra needed for the geometric construction. A monomial order below is degree followed by lexicographic order. It is a well-order and is preserved by multiplication. Every upward-closed subset of \(\mathbb N^n\) has finitely many minimal elements. Indeed any infinite sequence of nonnegative integers has an infinite nondecreasing subsequence: either some value repeats infinitely often or one successively chooses larger values. Applying this extraction to each coordinate proves that an infinite sequence in \(\mathbb N^n\) contains two terms comparable in their original order. Thus an infinite set of minimal elements is impossible. Every element lies above a minimal one, by descending the sum of coordinates.

For an ideal \(I\) in a polynomial ring over a field, choose polynomials whose leading monomials generate the upward-closed set of leading monomials of \(I\). Division by these finitely many polynomials terminates, since each subtraction decreases the leading monomial. The remainder of an element of \(I\) is still in \(I\), so a nonzero remainder would have a divisible leading monomial, a contradiction. This proves finite generation of every ideal and the monomial division criterion. Quotients and localizations are therefore Noetherian. In particular algebraic closed sets have finite irreducible decompositions: otherwise choose a minimal closed set without such a decomposition, using the descending-chain condition, and split it into two smaller closed sets. Both have finite decompositions, a contradiction.

Here is the normalization argument we need. If generators \(x_1,\ldots,x_n\) of a nonzero finite-type algebra satisfy a nonzero polynomial relation, choose an integer \(e\) larger than every exponent in that relation and replace \(x_i\), for \(i<n\), by \(y_i=x_i-x_n^{e^i}\). The highest power of \(x_n\) in the transformed relation has a nonzero constant coefficient: its exponents are the distinct base-\(e\) encodings \(a_n+\sum_{i<n}a_i e^i\). Dividing by that coefficient makes \(x_n\) integral over the subalgebra generated by the \(y_i\). Its finitely many lower powers generate the original algebra as a module over that subalgebra. Induction on the number of generators, stopping at an algebraically independent set, produces

For a domain \(A\), its fraction field is a finite extension of \(\mathbb C(t_1,\ldots,t_d)\). We use transcendence degree for \(d\). Its invariance under an algebraic field extension follows by exchange: clear denominators in an algebraic relation to solve for a parameter which occurs in it, replacing that parameter by the new transcendental element. Repeated exchanges compare two finite transcendence bases in both directions.

We recall and prove the required facts about integrality. If an algebra is generated by finitely many integral elements, the bounded powers in their monic equations generate it as a module. Conversely multiplication by an element on finitely many module generators gives a square matrix; multiplying \(tI-M\) by its adjugate gives its monic characteristic equation on that module. If the module contains \(1\), this is an equation for the element. These observations prove that integral elements form a ring and that integrality is transitive. Monic equations also show that integrality survives quotients and localization.

For an integral inclusion \(B\subset A\), every prime of \(B\) has a prime above it. Localize at that prime and choose a maximal ideal upstairs. Its contraction is maximal downstairs: a subring of a field over which that field is integral is a field, since a monic equation for the inverse of a nonzero element expresses that inverse in the subring. The local base has only one maximal ideal. This proves the assertion. Applying it after quotienting a prime proves going up. Two comparable primes above the same base prime are equal: quotient by the smaller and localize the base at its fraction field; the resulting integral domain algebraic over a field is a field. These are lying over, going up and incomparability in the form used here.

A field which is a finite-type \(\mathbb C\)-algebra is \(\mathbb C\). In (K.1) its polynomial subring must be a field by the preceding inverse argument, so \(d=0\); the field is then finite over \(\mathbb C\), and the fundamental theorem of algebra in E.1 applies. Hence maximal ideals of finite-type \(\mathbb C\)-algebras are their complex points. If \(f\) is not nilpotent in such an algebra, its localization at \(f\) is nonzero and is still of finite type; a maximal ideal there gives a complex point where \(f\ne0\). This proves the radical form of the Nullstellensatz as well. The prime-ideal and irreducible-closed-set descriptions consequently agree.

We will also use that polynomial rings over a field are unique-factorization domains and are integrally closed. Here is the algebra behind both assertions. For polynomials over a unique-factorization domain, factor out the greatest common divisor of the coefficients. The product of two primitive polynomials remains primitive: reduction modulo any prime factor of their contents would otherwise give a product of two nonzero polynomials equal to zero over a domain. Factorization over the fraction field, clearing denominators and using this content calculation, gives factorization and its uniqueness in the polynomial ring. Induct from the one-variable Euclidean algorithm. Finally, if a reduced fraction \(a/b\) satisfies a monic equation, every prime factor of \(b\) divides \(a\) after clearing denominators, a contradiction. Thus an integral fraction belongs to the ring.

If \(A\) is a domain finite over the polynomial ring \(B\) in (K.1), a nonzero \(g\in A\) has a monic minimal polynomial over \(\operatorname{Frac}B\) whose coefficients belong to \(B\). One way to see the coefficient assertion is to take the conjugate roots in a finite splitting field: they are integral over \(B\), so their symmetric expressions are integral, and those expressions lie in \(\operatorname{Frac}B\), where \(B\) is integrally closed. Its constant term is nonzero and belongs to \(gA\). Thus

For a proper irreducible closed subset of \(\operatorname{Spec}A\), take \(g\) in its nonzero prime ideal. Its coordinate algebra is finite over a quotient of \(B/(h)\). The images of the \(d\) parameters now satisfy a nonzero polynomial relation, so its transcendence degree is less than \(d\). This proves the strict dimension decrease used in all the finite inductions below.

We record explicitly the Chinese remainder calculation. For pairwise comaximal ideals \(I_1,\ldots,I_r\) of a ring \(R\), choose \(a_{ij}\in I_j\) with \(a_{ij}\equiv1\pmod{I_i}\) whenever \(i\ne j\). Then \(e_i=\prod_{j\ne i}a_{ij}\) is one modulo \(I_i\) and zero modulo every other ideal. Given residues represented by \(c_i\in R\), the element \(\sum_i c_i e_i\) maps to those residues. The kernel of this surjection \(R\to\prod_i R/I_i\) is exactly \(\bigcap_i I_i\). For two comaximal ideals, write \(a+b=1\) with \(a\in I\), \(b\in J\); if \(x\in I\cap J\), then \(x=ax+bx\in IJ\). Thus \(I\cap J=IJ\). Their powers are also comaximal: every term in \((a+b)^{2n-1}=1\) belongs to \(I^n\) or \(J^n\). In particular \((I\cap J)^n=I^n\cap J^n\), which will be used for the two node branches in L.2.

Finite morphisms of affine complex varieties are closed and have finite fibers. For closedness, quotient by the closed-set ideal and use lying over to identify the image with the zero set of its contraction. A fiber has a finite-dimensional coordinate algebra \(H\) over \(\mathbb C\). Any \(r\) distinct maximal ideals are pairwise comaximal, and their residue fields are \(\mathbb C\) by the Nullstellensatz just proved. The preceding surjection \(H\to\mathbb C^r\) forces \(r\leq\dim_{\mathbb C}H\), so there are finitely many such ideals. For later analytic use these morphisms are also proper in the usual topology. Each affine coordinate satisfies a monic equation over the base. Over a compact base set all its coefficients are bounded; a root of \(z^m+\sum_{j<m}a_jz^j\) has modulus at most \(1+\max|a_j|\). Indeed, for \(|z|>1+M\), where \(M=\max|a_j|\), the geometric sum gives \(\sum_{j<m}|a_jz^j|\leq M(|z|^m-1)/(|z|-1)<|z|^m\), contradicting the equation. The inverse image is therefore closed and bounded in a finite product of coordinate discs over that compact set, hence compact.

Every irreducible complex variety has a nonempty smooth algebraic open set. On an affine chart use (K.1). In characteristic zero its finite fraction-field extension is separable: an irreducible polynomial has no common factor with its nonzero derivative. It has a primitive element \(z\) which can be chosen as a complex linear combination of algebra generators. To justify this choice, the distinct embeddings in a splitting field are obtained by choosing roots successively in a tower of simple extensions; there are as many as the degree. Avoid the finitely many linear equalities which would give two embeddings the same value on \(z\). Such a choice exists over the infinite field \(\mathbb C\), for instance by substituting \((1,c,\ldots,c^{n-1})\) and avoiding the finitely many roots of the resulting nonzero polynomials. The number of conjugate values then equals the extension degree, so \(z\) generates it.

Let \(P(t,z)\) be its monic minimal polynomial. The other generators are polynomials in \(z\) with rational functions of \(t\) as coefficients. After inverting one nonzero polynomial \(h(t)\), \(A_h=B_h[z]=B_h[Z]/(P)\). Invert also the nonzero discriminant, or equivalently the denominator-cleared resultant of \(P\) and \(P_Z\). Their coprimality follows from separability; the Euclidean algorithm supplies the nonzero denominator-cleared identity. The analytic implicit-function argument of E.1, with several parameter variables and the same absolutely convergent series norm, identifies this open set with sheets over an open subset of \(\mathbb C^d\). It is smooth of dimension \(d\). The implicit derivative formulas also show that its tangent spaces depend algebraically on the point, using the nonvanishing Jacobian minors. This proves the dense smooth charts and algebraic tangent maps needed below.

K.2. Going down and an algebraic curve through a prescribed point

We need going down for a finite extension of domains \(B\subset A\), where \(B\) is an integrally closed domain of characteristic zero. Write \(K=\operatorname{Frac}B\), put \(\operatorname{Frac}A\) in a finite normal splitting field \(L/K\), and let \(C\) be the integral closure of \(B\) in \(L\). It is a ring by K.1, contains \(A\), and is integral over both \(B\) and \(A\). Let \(G\) be the finite Galois group. The fixed field is \(K\): an element outside \(K\) has at least two distinct conjugates by separability, and embeddings extend along the root tower in the splitting field. Thus \(C^G=B\), by normality of \(B\).

The group acts transitively on the primes of \(C\) over a given prime \(\mathfrak p\) of \(B\). Otherwise choose primes \(Q,P\) such that \(Q\) is none of the conjugates of \(P\). They are incomparable by K.1. Finite prime avoidance gives \(a\in Q\setminus\bigcup_{\sigma\in G}\sigma P\). Explicitly, for distinct incomparable primes \(P_i\), choose \(a_i\in Q\setminus P_i\) and \(b_i\in\bigcap_{j\ne i}P_j\setminus P_i\), the latter by multiplying elements of \(P_j\setminus P_i\); then \(\sum a_ib_i\) has the required property. But \(\prod_{\sigma\in G}\sigma a\) is in \(B\cap Q=\mathfrak p\), hence in \(P\), forcing some \(\sigma a\in P\), a contradiction.

Now let \(\mathfrak p_1\subset\mathfrak p_2\) in \(B\) and \(P_2\) in \(A\) lie over \(\mathfrak p_2\). Lift \(P_2\) to \(Q_2\) in \(C\) by lying over. First choose a prime of \(C\) above \(\mathfrak p_1\), then use going up to obtain \(Q'_1\subset Q'_2\) above the two base primes. A Galois automorphism carries \(Q'_2\) to \(Q_2\). Contracting its image of \(Q'_1\) to \(A\) gives

This proves going down without a finiteness assertion for \(C\).

Let \(Z\) be an irreducible affine variety, \(D\subsetneq Z\) closed, and \(p\in Z\). If \(p\in D\), there is an irreducible algebraic curve through \(p\) which is not contained in \(D\). Indeed choose (K.1) for its domain \(A\), a nonzero \(g\) vanishing on \(D\), and \(h\) as in (K.2). With \(a\) the image of \(p\) in \(\mathbb C^d\), choose a line through \(a\) on which \(h\) is not identically zero. Such a line exists by applying a nonzero homogeneous term of \(h(a+tv)\) to a direction \(v\). Here \(d\ge1\), since a zero-dimensional irreducible affine variety is a point by (K.1) and the Nullstellensatz.

Let \(\mathfrak q\) be the line ideal. Going down supplies \(P\subset\mathfrak m_p\) in \(A\) over \(\mathfrak q\). The domain \(A/P\) is finite over \(B/\mathfrak q\simeq\mathbb C[t]\); it defines a curve through \(p\). Since \(h\notin\mathfrak q\), (K.2) gives \(g\notin P\). Its intersection with \(D\) is finite: the zero set of \(g\) lies over the finitely many zeros of the nonzero polynomial \(h|_{\text{line}}\), and the finite morphism has finite fibers. For a point already outside \(D\), a constant arc will suffice in the next assertion.

K.3. The complex algebraic arc lemma

For the curve just obtained, there is a convergent analytic arc \(\gamma\) through \(p\) whose puncture lies outside \(D\). Here are the convergence and prescribed-endpoint details. Translate the finite coordinate \(t\) so that \(t(p)=0\). Choose a primitive element \(z\) of its function field as in K.1, additionally separating the finitely many points over \(t=0\). These add finitely many nonzero linear inequalities on its coefficients, so the same choice argument applies.

Its monic polynomial \(P(t,Z)\in\mathbb C[t,Z]\) is separable over \(\mathbb C(t)\). After inverting a polynomial \(h(t)\), every curve coordinate is a polynomial in \(z\) with coefficients in \(\mathbb C[t,h^{-1}]\). Choose a small punctured disc on which \(h\) and the discriminant have no zeros. The distinct roots of \(P\) form a finite analytic covering there, by E.1 and the monic root bound. Pulling back by \(t=u^m\), with \(m\) a multiple of the order of the permutation obtained by continuation around zero, makes each root single valued and analytic on the punctured \(u\)-disc. For clarity, the fundamental group used here is \(\mathbb Z\): lift a path's argument using finitely many short arcs; uniqueness gives its continuous lift to the logarithm half-plane. Loops lift with endpoint difference \(2\pi i n\), and two lifts with the same endpoints are homotopic by straight interpolation in that convex half-plane. Thus the power map kills precisely the needed finite monodromy.

A bounded analytic function on a punctured disc extends analytically across zero. This follows from the triangle-integral argument in E.1: apply it to \(f(\zeta)/(\zeta-u)\) on an annulus with a small circle around \(u\) removed. The circle around zero has integral tending to zero by boundedness; the circle around \(u\) tends to \(2\pi i f(u)\). The outer-circle formula for \(f(u)\) then expands into a power series by the uniformly convergent geometric series for \(1/(\zeta-u)\). This proves the asserted extension, not merely continuity.

All the root branches are bounded by their monic equation and hence extend. Every root of \(P(0,Z)\) is a limit of some branch. To verify that none is lost, take a sequence \(t_n\to0\), list all roots of each monic polynomial \(P(t_n,Z)\), and extract convergent subsequences using their common bound. Passing to the product of the linear factors gives \(P(0,Z)\), with multiplicities. There are only finitely many branches after the power pullback, so every such limit is one of their values at zero.

Choose a branch with \(z(0)=z(p)\). On its puncture the rational coordinate formulas give a point of the original curve. Each of these coordinates is itself integral over \(\mathbb C[t]\), so its monic equation bounds it too; each extends analytically. Their equations continue to hold at zero. The limit lies over \(t=0\) and has coordinate \(z(p)\), so it is \(p\), by the separating choice of \(z\). Shrink the disc to avoid the finite intersection with \(D\) away from \(p\). We have proved

For a reducible algebraic set and an open subset dense in every component, apply this proof to a component containing \(p\). Quasi-projective varieties reduce to an affine chart at \(p\). In particular a Zariski-dense open set in such a variety is dense in its usual topology as well. This is the only curve-selection statement used below.

K.4. Generic density in fibers

Let \(B\) be a finite-type complex domain and \(A=B[x_1,\ldots,x_n]/I\) a finite-type \(B\)-algebra. There is a nonzero \(b\in B\) such that \(A_b\) is a free \(B_b\)-module, possibly of infinite rank. To prove this particular generic-freeness statement, pass to \(K=\operatorname{Frac}B\). Choose a finite monic leading-monomial basis for \(I_K\) by K.1. Clear the finitely many denominators in its coefficients and its expressions in the original finite generators of \(I\). After inverting their product \(b\), these polynomials belong to \(I_b\) and are monic. Division of every original generator has zero remainder over \(K\), so it also has zero remainder over \(B_b\). Thus they generate \(I_b\). Division expresses the quotient in the monomials not divisible by their leading monomials. Those monomials are independent over \(K\), hence over \(B_b\). They give the asserted basis.

Suppose now that \(E\to Y\) is a dominant morphism of irreducible varieties and \(V\subset E\) is a dense open subset. There is a nonempty open \(Y^0\subset Y\) such that

Work first over an affine open \(Y=\operatorname{Spec}B\), and use a finite affine cover of \(E\). Each nonempty chart has domain \(A\), and dominance makes \(B\to A\) injective. Choose \(0\ne\delta\in A\) with \(D(\delta)\subset V\) on that chart. Apply the preceding argument to the finite-type algebra \(A/\delta A\). After inverting one base element it is free over the base. Consequently the exact sequence

splits as a sequence of base modules: lift its free basis. Reducing modulo any base maximal ideal therefore preserves its left injection. Hence \(\delta\) is a non-zero-divisor in every resulting fiber algebra.

It cannot lie in a minimal prime of that fiber. Localize there; the maximal ideal of this Noetherian zero-dimensional local ring is its nilradical and is nilpotent. The latter assertion follows by taking finitely many nilpotent generators and a power larger than the sum of their nilpotence exponents. A non-zero-divisor cannot become nilpotent in a nonzero localization. Thus \(D(\delta)\) is dense in every fiber component. This proves (K.5) on the chart; multiply the finitely many base elements for the affine cover. We used here the elementary description of the nilradical as the intersection of primes: a nonnilpotent element can be avoided by a prime, by choosing an ideal maximal among those disjoint from its powers and checking the prime property.

The same freeness argument shows that a dominant morphism of irreducible varieties has image containing a nonempty open subset of its target. Choose an affine source chart dominating an affine target and make its coordinate algebra free over the target by localization. It is nonzero, so each fiber is a nonzero finite-type complex algebra and has a complex point by K.1.

K.5. Generic Whitney regularity

Let \(T,S\subset\mathbb C^n\) be disjoint smooth irreducible locally closed algebraic sets. There is a nonempty algebraic open \(S^0\subset S\) such that \((T,S)\) satisfies Whitney (b) at every point of \(S^0\) in the closure of \(T\). If \(S\) is not contained in that closure, first remove the proper closed intersection; otherwise use the following construction.

For \(d=\dim_{\mathbb C}T\), take the graph

and let \(E\) be its reduced algebraic closure. The secant and tangent maps are algebraic; for the latter, solve the constant-rank Jacobian rows using a nonzero minor. The Grassmannian has finitely many algebraic matrix charts, obtained by putting a chosen invertible \(d\)-row minor equal to the identity. Their rational transition formulas give its algebraic structure; its Plücker coordinates are the minors. On a chart the usual quadratic Plücker equations solve every coordinate as one of these minors, so the decomposable locus is closed in projective space. In the usual topology it is compact, also seen from the closed bounded set of orthogonal projection matrices of rank \(d\).

The graph is open in \(E\): over \(T\times S\) its equations determine its two projective entries uniquely, and a graph of a regular map into a projective variety is closed there. The product \(T\times S\) is irreducible. For an algebraic verification on affine charts, if two nonzero functions in \(A\otimes_{\mathbb C}B\) had product zero, evaluation on each complex point of \(\operatorname{Spec}B\) would make one factor zero in the domain \(A\). Write each factor using linearly independent coefficients in \(A\); its zero-specialization locus is a proper closed subset of \(\operatorname{Spec}B\). Two such sets cannot cover an irreducible variety. Thus the tensor ring is a domain; the affine charts establish the assertion. It follows that \(E\) is irreducible and dominates \(S\).

Apply (K.5) to \(\Gamma\subset E\) and \(E\to S\), the second-coordinate projection. For \(p\in S^0\), every point of \(E_p\) is approachable by an analytic arc with puncture in \(\Gamma_p\), by (K.4). Now take any sequences \(x_i\in T\), \(y_i\in S\), tending to \(p\), with limiting secant line \(\ell\) and limiting tangent plane \(\tau\). Their graph points tend to \(q=(p,p,\ell,\tau)\in E_p\). Choose the just-proved arc through \(q\), whose second coordinate is identically \(p\). Its first coordinate has an expansion

The puncture belongs to \(T\), so \(x'(u)\in T_{x(u)}T\). After multiplication by \(u^{1-m}/m\), these tangent vectors tend to \(v\). The tangent planes tend to \(\tau\), hence \(v\in\tau\). The secant lines along the same arc tend to \([v]=\ell\). Therefore \(\ell\subset\tau\), proving (b).

This is also real Whitney (b) for the underlying smooth strata. Their tangent planes are complex subspaces regarded as real subspaces; a real limiting secant vector spans a complex line contained in the limiting complex tangent plane, so the real vector lies in the underlying real plane. Whitney (a) and the strict frontier dimension inequality follow as in I.2.

The assertion is intrinsic to an algebraic variety, not to an affine embedding. Two local embeddings can be compared with their common graph in the product of the ambient spaces. The coordinates of either embedding extend to regular, hence smooth, functions on a neighborhood in the other ambient space. The graph is carried to the zero-section embedding by the smooth triangular change \((u,v)\mapsto(u,v-g(u))\). Adding zero coordinates does not change tangent or secant inclusion, and I.2 proves invariance under this smooth coordinate change.

K.6. Finite compatible Whitney refinements

Every complex variety admits a finite algebraic Whitney stratification compatible with any prescribed finite family of constructible subsets. Here constructible means a finite union of algebraically locally closed subsets. Replace the family by the finitely many closed sets whose membership determines it, using a locally closed subset as the difference of its closure and a closed boundary.

We give a finite construction. At a stage let \(Z\) be the closed part not yet stratified, and retain the finitely many strata already chosen outside it. For each irreducible component \(C\) of \(Z\), choose a dense smooth open in one affine chart, by K.1. Delete its intersections with the other components. For each prescribed closed set and each closure of a previously chosen stratum, either it contains \(C\), or delete its proper intersection with \(C\). Thus membership in every such set will be constant on the new piece.

For each previous stratum \(T\) whose closure contains \(C\), apply K.5 to \(T\) and the chosen smooth open of \(C\), in the chart containing that open. Intersect the finitely many resulting nonempty opens. The result \(S_C\) is a nonempty smooth algebraic open in \(C\), satisfies (b) with every relevant previous stratum, and has all the required constant membership decisions. The pieces \(S_C\) are disjoint. Remove their union from \(Z\); its complement is closed, and every remaining component has smaller dimension by (K.2). Repetition therefore terminates after finitely many dimension decreases.

The frontier condition holds because every later piece was made either wholly contained in, or disjoint from, the closure of each earlier piece. A later closure cannot acquire a point in an earlier open piece: the later pieces lie in the closed residual set from that earlier stage. Every incident pair is (b)-regular by its construction when the lower piece was chosen. The construction preserved all prescribed membership conditions. This proves the theorem, including finiteness and the frontier condition.

Two variants will be used. First, any finite collection of finite algebraic stratifications has a common Whitney refinement: prescribe all their strata. A local system remains a local system on a refined smooth piece by inverse image. Second, if a closed subset \(Z\subset X\) remains to be stratified and a finite algebraic Whitney stratification of \(X\setminus Z\) is already fixed, the same construction extends it over \(Z\), without altering those fixed strata. Include their closures among the membership decisions and their pairs among the applications of K.5; if a fixed stratum has several irreducible components, test those finitely many components separately without changing the stratum. The previously fixed pairs remain unchanged. All these constructions work across a finite affine atlas by the embedding independence in K.5.

K.7. Compatible stratifications of an algebraic map

We first prove the needed algebraic critical-value assertion. A dominant map \(f:M\to N\) of smooth irreducible complex varieties has surjective differential on a nonempty open of \(M\). Choose a transcendence basis \(t_1,\ldots,t_e\) of \(\mathbb C(N)\), and extend it by \(u_1,\ldots,u_r\) to one of \(\mathbb C(M)\). Both remaining field extensions are finite separable. The derivations \(\partial/\partial t_i\) on the rational function field extend across a separable algebraic generator \(z\): differentiating its minimal equation gives

The nonzero denominator and the quotient polynomial rule verify both existence and uniqueness, and a tower gives the extension to the field. Thus the \(dt_i\) stay linearly independent in the source's rational differentials. Clear denominators in smooth coordinate charts from K.1. Some differential minor is nonzero, proving the assertion.

The critical locus in \(M\) is algebraically closed there, defined by the rank-drop minors in its finite smooth chart cover. No irreducible component \(R\) of it can dominate \(N\). Otherwise the dense smooth open of \(R\) would have generically surjective differential to \(N\) by the preceding argument, and its tangent vectors are tangent to \(M\); this contradicts criticality on \(M\). Thus the union of the closures of their images is a proper closed subset of \(N\). After deleting it and, if needed, the complement of the generic image supplied by K.4, the map is everywhere submersive and onto this target open. The same argument works simultaneously for finitely many smooth source pieces. Nondominant pieces are excluded by deleting their proper image closures.

Now let \(f:X\to Y\) be any algebraic morphism, with finite constructible markings in both spaces. We construct finite Whitney stratifications such that every source stratum maps onto a target stratum by a submersion. At a stage, let \(B\subset Y\) be the closed residual target and \(R=f^{-1}B\) the residual source; outside them keep the strata already fixed.

Extend the target stratification over \(B\) by K.6, respecting its markings, and choose its dense smooth pieces \(T_C\) on the irreducible components \(C\) of \(B\). Extend the source stratification over \(R\), respecting its markings and all \(f^{-1}T_C\). For each \(T_C\), apply the preceding critical-value argument to the finitely many source pieces mapping into it. Remove their critical-image closures and the image closures of nondominant pieces. For each dominant piece also restrict to the nonempty open contained in its image from K.4. Their intersection is a nonempty open \(T_C^0\subset T_C\). Its complement in \(C\) is closed and proper.

Retain the \(T_C^0\) and the source pieces above them. Every retained source piece is an open subset of a candidate Whitney stratum and is submersive and onto its target piece. These cuts are restrictions to the single open set \(Y\setminus B'\), and its inverse image, where \(B'=B\setminus\bigcup_C T_C^0\) is closed. Hence the retained pairs remain Whitney regular and keep the frontier incidences already constructed. Every component of \(B'\) has smaller dimension. Repeat the relative extension over \(B'\) and \(f^{-1}B'\). The process terminates with the asserted stratifications. Empty fibers cause no difficulty; they simply have no source stratum above them.

The smooth ambient extension required by J.7 is available locally. Extend the regular coordinate functions of \(f\) from an affine source embedding to its ambient open set. Near a target stratum, smooth submanifold coordinates from I.1 give a local retraction onto that stratum; compose with it. On the source part mapping to the stratum this equals \(f\). Thus J.7 applies to proper algebraic maps with the stratifications just constructed.

K.8. Constructibility of the classical operations

Let \(f:X\to Y\) be separated of finite type and \(F\) a bounded algebraically constructible complex of field sheaves. By K.6 its cohomology is constructible for a finite algebraic Whitney refinement. If \(f\) is proper in the classical topology, K.7 and J.7 show that \(Rf_*F=Rf_!F\) is bounded and algebraically constructible.

We include the ambient-space detail for a variety which has only local affine embeddings. Over a small coordinate box \(B\) in a target stratum, choose a slightly larger box with compact closure; the inverse image \(K\) of that closure is compact by properness. Cover \(K\) by finitely many source charts \(\psi_i\) which are locally closed algebraic embeddings in Euclidean spaces. Ambient smooth bumps, supported compactly inside these charts and positive on smaller sets covering \(K\), extend by zero and normalize to functions \(\chi_i\) with \(\sum\chi_i=1\) on a neighborhood of \(K\). Such bumps are those constructed in I.1. Regular changes of source coordinates extend locally to the ambient spaces, so these functions are smooth on every stratum and have compatible local ambient extensions.

The map \(z\mapsto(\chi_i(z),\chi_i(z)\psi_i(z))_i\), with zero entries off the respective supports, is an embedding near \(K\). Where \(\chi_i>0\), dividing its coordinate block by \(\chi_i\) recovers the original chart coordinates. This proves injectivity, a continuous local inverse and a smooth stratumwise immersion. In that chart, the remaining coordinates extend smoothly as functions of the recovered coordinates, so this is a graph embedding; the argument at the end of K.5 preserves Whitney regularity. Append the coordinate values of \(f\) to this embedding. Its image of \(K\) is compact, and its image of \(f^{-1}B\) is the intersection of that compact image with the ambient open set where the appended base coordinate lies in \(B\). It is closed in that ambient open set, and \(f\) is now the ambient coordinate projection. This reduces exactly to the ambient hypotheses of I.6 and J.7. The local topological properties used here follow from the finite affine atlas: local compactness and second countability hold in each chart, and separatedness makes the diagonal closed, hence makes the usual topology Hausdorff.

For proper supports of a general \(f\), work over an affine open \(V\subset Y\) and cover \(f^{-1}V\) by finitely many affine opens \(U_i\). Their finite intersections are affine: for two affine opens in a separated variety, the intersection is the inverse image of its closed diagonal in their affine product, hence is a closed affine subvariety of that product. Iterate. For any such intersection \(U\), embed it in \(\mathbb A^n\). The graph of \(f|_U\) is closed in \(V\times\mathbb A^n\); take its algebraic closure \(\overline U\) in \(V\times\mathbb P^n\). The original \(U\) is its open intersection with the affine chart, and

The properness follows directly from compactness of \(\mathbb P^n(\mathbb C)\): over a compact subset of \(V\), the closed set \(\overline U\) lies in its compact product with projective space. The extension \(j_!(F|_U)\) is algebraically constructible, with zero stalks on the algebraic boundary. K.6 and the proper case therefore prove that \(R(f|_U)_!(F|_U)=Rp_*j_!(F|_U)\) is bounded constructible.

For completeness, these finitely many affine computations recover \(Rf_!F\). The alternating complex of extensions by zero from the nonempty intersections \(U_{i_0}\cap\cdots\cap U_{i_q}\), augmented to \(F|_{f^{-1}V}\), is a resolution. At a point its stalk is the augmented simplex complex on the nonempty set of indices of opens containing that point, tensored with the stalk of \(F\). Inserting one fixed such index, with the alternating sign, contracts it. Totalize with a bounded representative of \(F\); the finite number of intersections keeps it bounded. Applying \(Rf_!\) and the composition isomorphism (H.8) expresses its result by finitely many triangles of the already treated pieces. Constructibility is closed under these triangles by Proposition 2.1 and common Whitney refinement K.6. Proper-support base change (H.6) identifies these computations with the restriction to \(V\). A finite affine cover of \(Y\), refined by its finite membership conditions, gives algebraic constructibility globally. The uniform degree bound is H.4.

Inverse image preserves bounded algebraic constructibility: refine the inverse images of a finite target stratification using K.6; on each resulting source stratum it pulls a local system back to a local system. Tensor and internal Hom preserve it by a common Whitney refinement and J.6. In particular \(D_XF\) is bounded algebraically constructible by (J.18). The just-proved proper-support assertion makes \(Rf_!D_XF\) constructible too, so both evaluation hypotheses in H.6 now hold. Its reversal gives

There is no hidden boundedness assumption on the original \(Rf_*F\) in this deduction. The pre-bidual identity of H.6 applied to \(D_XF\) already identifies \(D_YRf_!D_XF\) with \(Rf_*D_XD_XF=Rf_*F\); its left side is bounded. Evaluation then gives the first identity in (K.11).

Finally the pre-bidual pullback identity \(D_Xf^*=f^!D_Y\), applied to \(D_YG\), yields

for bounded algebraically constructible \(G\). The right side is bounded constructible, as just proved. Thus all six classical operations and internal Hom have the bounded algebraic constructibility asserted in Section 3, with the adjunctions, base change, projection and duality maps already constructed in H. These arguments use affine graph compactifications and finite open-cover descent; no unproved global compactification theorem is needed at this classical scope.

Monomial division, finite characteristic-zero going down, convergent prescribed-point arcs and generic graph-fiber density are the inputs to this complex-algebraic proof. For normalization, the Nullstellensatz, generic freeness and going down, see the Stacks Project references below. For generic Whitney regularity and finite stratifications, see [Kaloshin]. Real transversality is not required.

Appendix L. Nodes, normalization and the actual branches

We now justify the algebraic and analytic branch identifications used in Section 1 (the node calculation) and Exercise 3. A curve here is reduced and of pure dimension one. The local node argument works over an algebraically closed field of any characteristic. The analytic assertions and the general finite-normalization construction in L.3 are over \(\mathbb C\). In particular no topological comparison is used to assert an étale purity theorem.

L.1. Finite modules, completion and smooth curve points

We give the completion facts used below. Nakayama's lemma in the needed form says that a finite module \(M\) over a local ring \((R,\mathfrak m)\) with \(M=\mathfrak mM\) is zero. Choose generators and express this equality as \(v=Cv\) with matrix entries in \(\mathfrak m\). The adjugate of \(I-C\) shows that \(\det(I-C)\) annihilates every generator. This determinant is a unit since it is \(1\) modulo \(\mathfrak m\). Applying this to a quotient also shows that lifts of generators of \(M/\mathfrak mM\) generate \(M\).

For a Noetherian ring \(R\), every submodule of a finite module is finite. For a free module this follows by induction on its rank, projecting a submodule onto the last coordinate ideal and lifting its finite generators; pass to a free presentation for a general finite module. The polynomial ring \(R[T]\) is Noetherian as well. For an ideal, the ideals of possible leading coefficients in degrees at most \(d\) form an increasing sequence which stabilizes. Choose finitely many polynomials realizing generators up to a stabilization degree. Their multiples subtract the leading term of every sufficiently large-degree member of the ideal. The remaining bounded-degree part is a submodule of a finite free \(R\)-module, hence is finite. This proves the assertion, and induction handles finitely many variables.

Fix a Noetherian ring \(R\), an ideal \(I\), and finite modules \(N\subset M\). Put all ideal powers into the graded ring \(\mathcal R=\bigoplus_{n\geq0}I^nt^n\) and module \(\mathcal M=\bigoplus_{n\geq0}I^nMt^n\). Generators of \(I\), multiplied by \(t\), generate \(\mathcal R\) as an \(R\)-algebra; the polynomial-ring argument above makes it Noetherian. Generators of \(M\) in degree zero generate \(\mathcal M\). Thus its graded submodule \(\bigoplus_{n\geq0}(N\cap I^nM)t^n\) is finite. Replacing a finite generating family by its homogeneous components gives generators of degrees \(d_j\leq c\), for some \(c\). For \(n\geq c\), their degree-\(n\) multiples lie in \(I^{n-c}(N\cap I^cM)\), because \(I^{n-d_j}=I^{n-c}I^{c-d_j}\). The opposite inclusion follows by multiplication. We have proved Artin–Rees:

In particular \(I^nN\subseteq N\cap I^nM\subseteq I^{n-c}N\). These inclusions give canonical inverse maps between the limits for the two filtrations on \(N\): a coordinate modulo one filtration is obtained by projecting from a sufficiently fine coordinate of the other. Choosing a common finer index proves independence of the choice, compatibility and that the maps compose to the identity.

Write \(\widehat M=\varprojlim_{n\geq1}M/I^nM\). The finite-level sequence

is exact. Its kernel transition maps are surjective, since each kernel is a quotient of the same \(N\). This gives exactness on limits without assuming a general exactness theorem for inverse limits. For a compatible element in the right-hand limit, choose its lift at stage one. Having chosen a lift at stage \(n\), lift its prescribed right-hand value at stage \(n+1\) arbitrarily. The discrepancy at stage \(n\) lies in the kernel; lift that discrepancy through the surjective kernel transition and subtract it. The corrected lift is compatible. Induction proves surjectivity on limits. Injectivity and equality with the kernel are checked coordinate by coordinate. The cofinal-filtration comparison now identifies the left-hand limit with \(\widehat N\). Hence completion is exact on finite modules.

A finite free presentation \(R^a\to R^b\to M\to0\) yields the same cokernel after completion and after tensoring with \(\widehat R\): both maps of finite free \(\widehat R\)-modules have the original matrix entries. Right exactness of tensor follows directly from its balanced-map universal property, which imposes precisely these relations. Thus the natural tensor map is an isomorphism. Applying exactness to \(I^n\subset R\) gives the second identity below: the quotient \(R/I^n\) is already complete, and a finite set of generators of \(I^n\) shows that the image of its completion is \(I^n\widehat R\).

The same argument applied to \(I^nM\subset M\), using the surjection from a finite direct sum of copies of \(M\) defined by generators of \(I^n\), gives \(\widehat M/I^n\widehat M=M/I^nM\). These identifications commute with the quotient transition maps. Consequently \(\widehat M\) is complete and separated for its own ideal-power topology, not just for an unspecified limit topology.

For separation before completion, put \(N=\bigcap_n I^nM\). It is a finite submodule. Formula (L.1) at \(n=c+1\) gives \(N=IN\). The determinant proof of Nakayama gives an annihilator of \(N\) in \(1+I\). If \(R\) is local and \(I\) is contained in its maximal ideal, this annihilator is a unit, so \(N=0\). Thus \(M\to\widehat M\) is injective for every such ideal, and \(\widehat M=0\) implies \(M=0\). Applied to a quotient, the same assertion says that every submodule of a finite module is closed for this ideal-power topology. In particular it applies to the ideal \(\mathfrak mB\subset\mathfrak n\) in U.2, which need not equal \(\mathfrak n\). The complete filtration proof is also given in Noetherian and Artinian rings, Sections 5–6; compare [Stacks, finite-module completion].

We record two further properties of the completed ring used with formal local equations. If \(I=\mathfrak m\) is maximal, an element of \(\widehat R\) with nonzero residue in \(R/\mathfrak m\) is a unit: its coordinates are units in every \(R/\mathfrak m^n\), and their inverses are compatible. The projection to \(R/\mathfrak m\) is surjective by successive lifting, so \(\widehat R\) is local with maximal ideal \(\mathfrak m\widehat R\), by (L.2).

For any ideal \(I\), the ring \(\widehat R\) is Noetherian. Set \(J=I\widehat R\). The consecutive quotient identities in (L.2) identify \(\operatorname{gr}_J\widehat R\) with \(\operatorname{gr}_I R\), a quotient of a polynomial ring over \(R/I\) on finitely many degree-one generators. This graded ring is therefore Noetherian. For an ideal \(Q\subset\widehat R\), its initial classes form a homogeneous ideal in this graded ring. Choose finitely many homogeneous generators and lift them to \(q_1,\ldots,q_s\in Q\), of orders \(d_i\). Starting with \(q\in Q\), cancel the degree-\(n\) initial class of the current residual by a sum \(\sum_i a_{i,n}q_i\), where \(a_{i,n}\in J^{n-d_i}\), using zero when \(d_i>n\). The residual then lies in \(Q\cap J^{n+1}\). Each coefficient sum \(a_i=\sum_n a_{i,n}\) converges by completeness, since the orders tend to infinity. Separatedness makes the limiting residual zero, so \(q=\sum_i a_iq_i\). Thus \(Q\) is finitely generated. No prior closedness assumption on \(Q\) entered this argument. These completion statements are also proved in Completion, Sections 1–3.

For a local finite-type \(k\)-algebra at a \(k\)-rational point, translate the point to the origin in a polynomial presentation. Modulo the \(n\)-th power of its maximal ideal, every polynomial inverted in forming the local ring is already a unit: its constant term is nonzero, and a finite geometric series in its nilpotent remaining part gives an inverse. Hence the completed ambient polynomial local ring is \(k[[z_1,\ldots,z_r]]\), coefficient by coefficient. Exactness of completion and a finite generating family for the equation ideal give its quotient by those same equations. Finally a power-series coordinate map with invertible linear part has a unique inverse: after degrees below \(d\) have been determined, cancel the degree-\(d\) error by that invertible linear matrix. Induction gives compatible inverse maps at every order, hence inverse power-series maps. Consequently \(r-1\) equations with independent linear terms have completed local ring \(k[[z]]\), where \(z\) is any complementary linear coordinate.

We will need two elementary valuation facts. Suppose a Noetherian local ring has maximal ideal \((z)\), with \(z\) a non-zero-divisor and not a unit. The intersection \(J=\bigcap_n z^nR\) satisfies \(J=zJ\): divide an element by \(z\) and cancel \(z\) in every higher divisibility statement. Nakayama gives \(J=0\). Every nonzero element is therefore uniquely \(z^e u\), with \(e\ge0\) and \(u\) a unit. Products add exponents, so the ring is a domain. Fractions of negative exponent cannot satisfy a monic equation, whose lowest exponent would occur only in its leading power. Thus this is a discrete valuation ring and is integrally closed.

Conversely a Noetherian normal local domain \(R\) of dimension one is such a ring. Choose \(0\ne a\in\mathfrak m\). Every nonzero prime is maximal, so \(\sqrt{(a)}=\mathfrak m\), and finite generation gives \(\mathfrak m^n\subset(a)\) for some least \(n\ge1\). Choose \(b\in\mathfrak m^{n-1}\setminus(a)\) and set \(w=b/a\). Then \(w\notin R\) but \(w\mathfrak m\subset R\). If \(w\mathfrak m\subset\mathfrak m\), the adjugate argument on generators of the faithful module \(\mathfrak m\) makes \(w\) integral, a contradiction. Hence \(wc\) is a unit for some \(c\in\mathfrak m\). Rescale so that \(wc=1\). Every \(d\in\mathfrak m\) equals \(c(wd)\), proving \(\mathfrak m=(c)\).

A one-dimensional finite-type local \(k\)-algebra with residue field \(k\) and maximal ideal generated by a non-zero-divisor is a smooth curve point. Here is the required presentation argument. Its tangent space has dimension one. In an affine embedding, choose \(n-1\) equations with independent linear terms and let \(S\) be their local quotient. Its completed ring is \(k[[z]]\) by the formal inverse construction, and \(S\) injects into it by (L.2). Its maximal ideal is \((z)\) by Nakayama, so it is a discrete valuation ring by the preceding argument. The given local ring is \(S/J\) and has dimension one. Any nonzero ideal of this discrete valuation ring contains a power of \(z\) and has zero-dimensional quotient. Thus \(J=0\). The chosen Jacobian minor stays invertible on an algebraic neighborhood. This supplies the local smooth presentation. Over \(\mathbb C\), E.1's convergent implicit-function argument makes it an actual analytic disc, with \(z\) an analytic coordinate. No formal-to-convergent approximation theorem is being assumed.

L.2. The node criterion and its finite normalization

Let \(A=\mathcal O_{X,p}\) for a reduced curve over an algebraically closed field \(k\), with maximal ideal \(\mathfrak m\) and residue field \(k\). The following two conditions are equivalent:

1. \(\widehat A\simeq k[[u,v]]/(uv)\).
2. \(\dim_k\mathfrak m/\mathfrak m^2=2\), and the kernel of \(\operatorname{Sym}^2_k(\mathfrak m/\mathfrak m^2)\to\mathfrak m^2/\mathfrak m^3\) contains a product of two distinct linear forms.

The implication from 1 to 2 follows from (L.2) and the degree-two relation \(uv\). We prove the converse while constructing the normalization.

Choose a basis \(x,y\) of \(\mathfrak m/\mathfrak m^2\), lifted to \(A\), such that the coefficient of \(y^2\) in the given quadratic relation is nonzero and \(x\) is a non-zero-divisor. These requirements can be met simultaneously. A reduced Noetherian ring embeds in the product of the fraction fields of its finitely many minimal-prime quotients: their intersection is its zero nilradical. Avoiding every minimal prime therefore makes an element a non-zero-divisor. In our one-dimensional local curve none of these primes is maximal. Since lifts of a tangent basis generate \(\mathfrak m\), the linear combinations lying in any such prime form a proper linear subspace of the coefficient space. The two forbidden tangent directions impose two further proper subspaces. A finite union of proper linear subspaces cannot cover a vector space over the infinite field \(k\): choose a polynomial line avoiding their finitely many nonzero linear equations. This gives \(x\) and then a complementary \(y\).

Scale the quadratic relation to obtain \(y^2-a_0xy-b_0x^2\in\mathfrak m^3\), where \(T^2-a_0T-b_0\) has two distinct roots in \(k\). Since \(\mathfrak m=(x,y)\), the relation gives \(\mathfrak m^2=x\mathfrak m+\mathfrak m^3\). Nakayama on \(\mathfrak m^2/x\mathfrak m\) gives \(\mathfrak m^2=x\mathfrak m\). Now \(\mathfrak m^3=x^2\mathfrak m\), so the error can be absorbed into the coefficient of \(x^2\) without changing its residue. Thus for some \(a,b\in A\), reducing to \(a_0,b_0\),

The ring \(B=A+At\) is finite over \(A\), is reduced, and is birational: after inverting \(x\) it is \(A[1/x]\). Moreover \(\mathfrak mB=xB\) and \(\mathfrak m B\subset A\), because \(xt=y\) and \(yt=ay+bx\).

There is a surjection \(k[T]/(T^2-a_0T-b_0)\twoheadrightarrow B/\mathfrak mB\). Its target is nonzero by lying over, proved in K.1 using only integrality. It cannot have dimension one. If it did, the constants from \(A\) would surject onto it, so \(B=A+\mathfrak mB\); Nakayama on the finite module \(B/A\) would imply \(B=A\). Then \(t\in A\) and \(y=xt\), making \(\mathfrak m=(x)\), contrary to its tangent dimension two. Therefore the surjection is an isomorphism, and

The two maximal ideals \(\mathfrak n_1,\mathfrak n_2\) of \(B\) above \(\mathfrak m\) are all its maximal ideals, since \(B\) is integral over the local ring \(A\). Each localization has maximal ideal generated by \(x\), which remains a non-zero-divisor. L.1 makes each a discrete valuation ring. Hence \(B\) is normal in its total quotient ring: an integral fraction belongs to every maximal localization, and a fraction belonging to all of them belongs to \(B\). For the latter assertion, its denominator ideal \(\{c\in B:cq\in B\}\) would otherwise be a proper ideal contained in a maximal ideal where the fraction did not belong. Thus \(B\) is the integral closure of \(A\). Indeed anything integral over \(A\) is integral over \(B\) by the same monic equation, while every element of \(B\) is integral over \(A\).

The residue difference in (L.4) gives an exact sequence of finite \(A\)-modules

To check its kernel, \(\mathfrak m\) kills \(B/A\), which is generated by the class of \(t\). That class is nonzero, so \(B/A\) has dimension one. Its two residues differ by the nonzero difference of the two roots of the quadratic. This proves exactness, including the specified last map.

Complete (L.5) using L.1. Since \(\mathfrak mB=xB=\mathfrak n_1\cap\mathfrak n_2\) and the two maximal ideals are comaximal, the Chinese remainder calculation in K.1 applies at every power and gives \(B/x^nB=B/\mathfrak n_1^n\times B/\mathfrak n_2^n\). Localization at \(\mathfrak n_i\) does not change its quotient: every element outside that ideal is invertible modulo its power by a finite geometric series. In its discrete valuation ring, successive subtraction of residue constants gives unique expansions modulo \(x^n\). Thus

The last isomorphism sends \(u\) to \((x,0)\) and \(v\) to \((0,x)\). Every pair with equal constant terms has a unique constant plus a positive-power series in each variable. This proves the node criterion, its two actual normalization points, their regular local rings and the completed comparison. The argument also covers characteristic two: the only needed quadratic condition is two distinct linear factors.

This is a criterion on the algebraic local ring, not a definition requiring a prior plane presentation. In particular a reduced plane curve with a nonzero quadratic term which is a product of distinct lines is nodal. Its tangent space has dimension two, and that quadratic term gives condition 2. Conversely a completed node has exactly that tangent relation. A finite presentation of (L.3) spreads the local normalization to a finite map on an algebraic neighborhood of \(p\): clear its finitely many denominators outside \(\mathfrak m\). The smooth presentations of the two local rings from L.1 hold on neighborhoods of both inverse-image points. Their complement has closed image under the finite map by K.1; remove that image from the base. On the resulting neighborhood the finite ring is normal and birational, and hence is its normalization by the same integral-closure argument.

L.3. Finite normalization of a complex curve

For the multiple-branch comparison we also need existence of the algebraic normalization beyond nodes. We prove it at the complex curve scope. Let \(A\) be the coordinate domain of an affine irreducible complex curve. K.1 supplies a finite inclusion \(R=\mathbb C[s]\subset A\). Its fraction field \(L\) is finite separable over \(K=\mathbb C(s)\). Choose the primitive element \(z\in A\) as in K.1; \(e_i=z^i\), \(0\le i<d=[L:K]\), form a \(K\)-basis.

Here is the trace calculation, including its nondegeneracy. In a splitting field let \(\sigma_1,\ldots,\sigma_d\) be the distinct embeddings of \(L\), constructed by the root choices in K.1, and put \(V_{ri}=\sigma_r(z)^i\). Its determinant is the Vandermonde product: it vanishes when two row parameters coincide, has the total degree of the product of their pairwise differences, and the coefficient of the diagonal monomial fixes the remaining constant as one. The conjugate values of \(z\) are distinct, so \(V\) is invertible. If \(M(c)\) is the matrix of multiplication by \(c\in L\) in the basis \(e_i\), evaluating each column gives

The second identity follows by conjugation and \(\operatorname{tr}(UV)=\operatorname{tr}(VU)\), obtained by interchanging the two finite summation indices. Consequently the trace pairing matrix \(G_{ij}=\operatorname{Tr}_{L/K}(e_ie_j)\) equals \(V^{\mathsf T}V\), and its determinant \(\Delta\) is the square of the nonzero Vandermonde product. Its entries lie in \(K\), as traces of matrices over \(K\). Each \(\sigma_r(z)\) is integral over \(R\), because it satisfies the same monic equation as \(z\). The trace sum and K.1's closure of integral elements under sums and products show that every entry of \(G\) is integral over \(R\). Since \(R\) is integrally closed, \(G_{ij}\in R\) and \(0\ne\Delta\in R\).

Let \(C\) be the integral closure of \(A\) in \(L\). Its elements are integral over \(R\) by transitivity. For \(c=\sum c_i e_i\in C\), every trace \(\operatorname{Tr}(ce_j)\) is likewise integral over \(R\) and belongs to \(R\). Solving the trace matrix equation gives \(c_i\in\Delta^{-1}R\). Hence

It is an \(R\)-submodule of this finite module, so is finite. The same generators make it finite over \(A\). It is normal: an element integral over \(C\) is integral over \(A\) by transitivity and therefore lies in \(C\).

For a reduced affine curve, apply this to its finitely many irreducible component domains. The product of their normalizations is a finite algebra over \(A\): each component quotient is a cyclic \(A\)-module, and each normalization is finite over that quotient. It is integral, embeds in the product of the component fraction fields and contains every element of that product integral over \(A\), by the domain case. This is precisely the integral closure in the total quotient ring. To see that this total quotient ring is the indicated product, choose an element in every other minimal prime but outside one specified minimal prime, using finite prime avoidance as in K.2. Sums of these elements have nonzero image in every component and can be inverted; their ratios give the component idempotents. Each component's nonzero denominator can then be included in such a sum, giving its full fraction field.

Integral closure commutes with localization here. If a fraction is integral after localizing, one denominator from the multiplicative set clears all coefficients of its monic equation. A sufficiently large power of that denominator times the fraction then satisfies a monic equation over the original ring. This proves both inclusions. The affine constructions consequently glue to a finite normalization \(\nu:\widetilde X\to X\). A finite morphism is closed algebraically by K.1 and is proper in the usual topology by the same monic root bounds used there.

The normal local rings of \(\widetilde X\) at its closed points are one-dimensional: their component rings are finite over \(\mathbb C[s]\), and lying over, going up and incomparability give this dimension. L.1 makes them discrete valuation rings and then actual smooth analytic curve points. Thus \(\widetilde X\) is a smooth complex curve. On each affine chart finitely many generators of its normalization are fractions on the original components. A common denominator nonzero on every component gives \(A_d=C_d\). Its zero set on a curve is finite, by the strict dimension decrease in K.1. Therefore \(\nu\) is an isomorphism away from finitely many points locally on \(X\). At a smooth point the local ring is already a discrete valuation ring, so the normalization is an isomorphism there as well.

L.4. The convergent branches of an algebraic node

Return to a complex node \(p\). In (L.3), \(x\) is a uniformizer at each of the two points \(q_i\) above \(p\). By L.1 it is an analytic coordinate on a disc at \(q_i\), and the regular function \(t\) has a convergent expression \(t_i(x)\) there. Their constant terms are the two distinct roots \(\lambda_i\) in (L.4). Hence the normalization maps these discs to

To interpret these as full ambient branches, choose an affine embedding of \(X\) near \(p\). Tangent dimension two gives \(n-2\) equations with independent linear terms. Keep \(x,y\) as the complementary coordinates; these regular functions extend to an ambient neighborhood. Let \(S\) be the local ring cut out by those equations and write \(A=S/J\). The formal inverse construction gives \(\widehat S=k[[x,y]]\). Its surjection to the completed node has principal kernel: lift the two node parameters to \(\widehat S\); their independent linear terms make them formal coordinates, and in those coordinates the kernel is their product. Exact completion and reduction modulo the maximal ideal now give \(\dim_k J/\mathfrak m_SJ=1\). Nakayama supplies a single generator \(f\) of \(J\), whose initial term is the nondegenerate quadratic relation. These local equation equalities hold on a sufficiently small algebraic neighborhood after clearing the finitely many denominators.

The convergent implicit-function argument of E.1 eliminates the other coordinates, so the analytic curve is cut out in that surface by the convergent function \(f(x,y)\). Its restriction to each graph in (L.8) is zero. Division by \(y-x t_1(x)\) is convergent: the identity \(f(x,y)-f(x,g(x))=(y-g(x))\int_0^1\partial_y f(x,g(x)+s(y-g(x)))\,ds\) gives a convergent analytic quotient on a smaller polydisc. On the other graph the quotient vanishes for \(x\ne0\), hence also at zero by continuity. A second division therefore gives \(f=(y-x t_1(x))(y-x t_2(x))h\), with \(h(0)\ne0\) by the quadratic initial term. Thus the following coordinate change identifies the analytic equation itself, as well as its underlying set.

There are no additional points of \(X\) near \(p\) outside these images. The normalization is surjective by lying over. Its fiber at \(p\) consists of \(q_1,q_2\), and properness allows the neighborhood to be shrunk so that its full inverse image lies inside the two chosen discs. Explicitly, the complement of their union is closed and has closed image under a proper map; that image avoids \(p\). Proper maps here are closed in the usual topology: a convergent sequence in such an image lies in a compact base set together with its limit, and properness provides a convergent subsequence of inverse-image points. Thus the two graphs in (L.8) describe exactly the local set.

The analytic coordinate change

has nonzero Jacobian determinant \(\lambda_2-\lambda_1\) at zero. Its inverse is convergent by E.1. It sends the local curve to \(uv=0\), and its two normalization discs to the two coordinate branches. In particular this proves the asserted analytic local model from the completed-local-ring criterion, without replacing an algebraic normalization by a merely formal one. Each branch has its usual disc topology; their union has the quotient topology proved in B.4. At a node the normalization pushforward is therefore the sheaf on those two separated discs used in Proposition 1.3.

L.5. Several smooth branches and the normalization sheaf

Suppose a reduced complex algebraic curve has, near a point \(p\), exactly \(r\) smooth analytic branches meeting only at \(p\). Distinct tangent lines, as in Exercise 3, imply that last property after shrinking by B.4. The finite normalization of L.3 separates exactly these \(r\) branches and is locally an isomorphism from a disc onto each of them.

Here are the comparison details. The inverse-image set of \(p\) is finite. Each of its points has a smooth disc neighborhood by L.1, and these neighborhoods can be chosen disjoint. Remove the finitely many other exceptional image points from a small neighborhood of \(p\); there the normalization is an isomorphism off \(p\). Properness, as in L.4, then keeps the full inverse image inside the chosen discs. Each sufficiently small punctured disc is connected and maps into one of the disjoint punctured branches. In that branch's analytic parameter its map is a nonconstant holomorphic function vanishing only at the centre, with expression \(h(z)=z^e a(z)\), where \(a(0)\ne0\).

The function \(a(z)\) has a convergent \(e\)-th root near zero: divide by its constant term, choose a root of that constant, and apply the binomial series near \(1\). Its convergence follows from the ratio test and its identity from multiplication of power series. Thus a local coordinate change puts \(h\) in the form \(w\mapsto w^e\). Since the normalization is one-to-one off \(p\), \(e=1\). So the disc maps isomorphically onto a neighborhood in its smooth branch. Two different inverse-image discs cannot map to the same branch, again by injectivity off \(p\). Every branch is reached, by surjectivity and properness. This proves the required bijection and the local isomorphisms.

Consequently, with \(i_p:\{p\}\hookrightarrow X\), the actual algebraic normalization gives the local exact sequence of classical sheaves

Away from \(p\) it is the identity sequence, and at \(p\) it is the diagonal inclusion, so exactness is stalkwise. For \(r=2\), choosing an order of the branches identifies its final map with their difference; exchanging them changes this identification by a sign. For several singular points the global quotient is the sum of these local quotients. The branch labels carry no preferred global order. B.4 now applies to the actual normalization in Exercise 3, and G.4 applies to the actual nodal normalization in Section 1 (the node calculation), with their previously proved support complexes and duality maps.

There is also a purely algebraic comparison under the étale smooth-branch hypothesis in Exercise 3. If the branches in the chosen neighborhood have prime ideals \(P_1,\ldots,P_r\), their local ring product \(\prod_i A/P_i\) is finite over \(A\), since each quotient is cyclic. Its factors are the regular local rings of the given smooth curves, hence are discrete valuation rings by L.1. It embeds in the total quotient ring of \(A\), which is the product of the component fraction fields as proved in L.3 by prime avoidance; that argument works over any field. The product is normal and integral over \(A\), so it is exactly the local normalization. It separates the \(r\) branch points. This verifies the normalization assertion under the stated smooth-branch hypothesis, without asserting that every singularity has such branches.

At a geometric node over an algebraically closed field of positive characteristic, L.2 supplies the finite normalization with its two smooth algebraic branch points and the module comparison (L.5) directly, without a prior split-branch presentation. Thus the local branch geometry for the étale nodal computation does not require a classical analytic model. The sheaf computation and its descent from geometric branch neighborhoods still use the separately identified finite-map étale base change and smooth purity arguments in Section 3. Over a finite field branch labels may be exchanged by Frobenius; the displayed two copies in geometric stalk and costalk groups do not assert that each summand is arithmetically fixed. Singular finite-coefficient duality and the normalized adic passage are treated in M and BV–BX.

For nodes, completion, Artin–Rees and discrete valuation rings, see the Stacks Project references below. The finite ring \(A[y/x]\), its two residue maps and the completion sequence give the normalization and convergent branch identification of L.2–L.4.

Appendix M. Ordinary étale finiteness and singular biduality over a field

We prove the field case needed here, including the steps which a reference to a general finiteness theorem would leave unwritten. Throughout this appendix, schemes are separated and of finite type over a field \(k\), \(n\) is invertible in \(k\), and \(\Lambda=\mathbf Z/n\). Constructibility means finite locally constant values on a finite locally closed partition; it does not require the pieces to be smooth over an imperfect field. The ordinary direct-image assertion also holds for any finite coefficient ring whose order is invertible in \(k\), by forgetting to a \(\mathbf Z/n\)-module. The duality assertion below uses the self-injectivity of \(\mathbf Z/n\), not a claim that every finite ring is self-injective.

The proofs use the finite-coefficient proper-support finiteness and smooth trace of Propositions 3.2–3.3. Their earlier-programme inputs are the actual proper and smooth base-change theorems identified in Section 3; the finite presentations and Noetherian subobject argument in Constructible sheaves and extension by zero, §§4–7; and the cohomological-dimension and continuity arguments in Cohomological dimension and the Künneth formula, §§1–8. Proposition 3.3 constructs the exceptional adjoint by proving representability through presheaves and exact sheafification, with exact programme locators for compact representability and unbounded resolutions. Its internal-Hom adjunction and local description are used as written in Poincaré duality for smooth varieties, §§8–10. These arguments use those geometric and categorical hypotheses at the stated coefficient scope.

For finiteness and duality, see the author edition of SGA \(4\frac12\), Théorèmes de finitude, §§2.1–2.8 and 4.1–4.7. We give a field-only dimension argument below; no generic base-change theorem over an arbitrary parameter scheme is assumed.

M.1. Preparations: fields, smooth open subsets and bounds

First, ordinary higher images have a finite bound before their constructibility is known. On an affine target and an affine source of dimension at most \(d\), the support estimate of the cohomological-dimension companion, Theorem 6.2, gives \(R^qf_*F=0\) for \(q>d\), for every torsion sheaf \(F\). Cover a general separated source by \(r\) affine opens. All multiple intersections are affine, of dimension at most \(d\). The finite Čech-to-derived-image spectral sequence therefore gives the sufficient bound \(d+r-1\). Finite affine covering of the target gives a uniform bound. This is a bound on the relative functor, so it does not assume that \(k\) has finite Galois cohomological dimension. The same companion proves filtered-colimit continuity and, over a separably closed field, absolute torsion cohomological dimension at most \(2d\). Every spectral sequence below consequently has a finite contributing range when the input is bounded.

Finite-cover bound (M.1a). Let \(f:X\to Y\), and let \(X_1,\ldots,X_r\) be a finite open cover, with \(r\geq1\). For a nonempty increasing tuple \(J=(i_0,\ldots,i_p)\), put \(X_J=\bigcap_{i\in J}X_i\), let \(v_J:X_J\to X\) be the inclusion, and put \(f_J=fv_J\). Suppose, for the coefficient category in question, every \(f_J\) has higher-image bound \(D\geq0\). Then \[ R^nf_*F=0\quad(n>D+r-1) \tag{M.FC1} \] for every sheaf \(F\) in that category. More generally, if \(H^qK=0\) outside \([a,b]\), then \[ H^n(Rf_*K)=0\quad(n<a\text{ or }n>b+D+r-1). \tag{M.FC2} \]

Here is a construction of the comparison and its bound. Work first with abelian étale sheaves or with \(\Lambda\)-module sheaves. Choose an injective resolution \(F\to I^\bullet\), zero in negative degrees. For \(0\leq p<r\), set \[ C^{p,q}= \bigoplus_{i_0<\cdots<i_p} v_{J*}(I^q|_{X_J}),\qquad (\delta c)_{i_0\ldots i_{p+1}} =\sum_{\nu=0}^{p+1}(-1)^\nu c_{i_0\ldots\widehat{i_\nu}\ldots i_{p+1}}|_{X_J}. \tag{M.FC3} \] An empty intersection contributes zero. The horizontal and vertical maps commute, so the total differential on \(C^{p,q}\) is \(\delta+(-1)^p d_I\); its square is zero. The augmentation \(I^q\to C^{0,q}\) is restriction.

Every augmented horizontal row is exact. On an étale test object mapping into \(X_t\), treat a tuple of indices as alternating, with value zero when an index repeats. Define \(hc\) by inserting \(t\) as the first index. This is a map on sections there because intersection with \(X_t\) changes none of that test object's sections. In \(\delta h+h\delta\), deleting the inserted index leaves \(c\); deletion of each other index occurs twice with opposite signs. Thus \(\delta h+h\delta=1\), including the augmented degree, where \(hc=c_t\). These test objects cover \(X\), proving exactness as sheaves.

The augmentation \[ I^\bullet\longrightarrow\operatorname{Tot}C^{\bullet,\bullet} \tag{M.FC4} \] is a quasi-isomorphism. To see this directly, include the augmented column in the total complex. In a total cocycle take its component of smallest vertical degree. Its horizontal differential is zero, so row exactness lets us subtract a total boundary removing it. The new component has vertical degree one greater and horizontal degree one less. There are only \(r+1\) augmented columns. At the last column the augmentation is injective, so the remaining component is zero. This proves the augmented total is acyclic.

The terms of \(\operatorname{Tot}C\) are injective. Indeed restriction \(v_J^*\) has the exact left adjoint \(v_{J!}\), and \(v_{J*}\) has the exact left adjoint \(v_J^*\). Applying either adjunction to a monomorphism proves that these right adjoints preserve injectives. The open adjunction and extension construction are proved in BB.3: restriction keeps the geometric stalk, and extension by zero keeps it on the open and gives zero outside. Thus both functors are exact. BB.3 also supplies enough module injectives; iterating embeddings and cokernels gives the bounded-below resolution used here. Finite sums of injectives are injective by applying Hom componentwise. Consequently (M.FC4) is a comparison of injective resolutions, and applying \(f_*\) computes the actual \(Rf_*F\) by \[ T^{p,q}=\bigoplus_{|J|=p+1}f_{J*}(I^q|_{X_J}). \tag{M.FC5} \] The comparison comes from restriction maps and the chosen resolution; it is not an identification chosen separately on cohomology.

Filter this total complex by columns numbered at least \(p\). Its successive quotient is the \(p\)-th column placed in total degrees \(p+q\). The restriction \(I^\bullet|_{X_J}\) is an injective resolution of \(F|_{X_J}\), so that quotient has cohomology \(\bigoplus_{|J|=p+1}R^qf_{J*}(F|_{X_J})\) in degree \(p+q\). It vanishes above \(D+p\). Starting with the last column, the long exact sequences of these finitely many filtered subcomplexes give the upper bound \(D+r-1\), proving (M.FC1). There is no negative cohomology. The finite ordinary truncation tower of \(K\) has successive terms \(H^q(K)[-q]\), each with image in degrees \([q,q+D+r-1]\). The triangle cohomology sequences give (M.FC2).

For the geometric application, make \(Y\) affine and cover its inverse image by \(r\) affine opens. Separatedness over \(k\) makes their intersections affine: a twofold intersection is the inverse image of the closed diagonal in the product of the two affine opens, and further intersections follow by induction. Each \(f_J\) is therefore affine. Theorem 6.2 of the cohomological-dimension companion gives \(D=d\), where \(d\) bounds the dimensions of these opens. In a finite module category the underlying-cohomology comparison of BB.3 gives the same bound. Restriction to an open in \(Y\) commutes with this derived image: sections agree before deriving, and restriction to the inverse-image open preserves injectives by the adjunction just proved. A finite affine cover of \(Y\) therefore gives one bound by taking the maximum of its finitely many \(d+r-1\). The empty source has zero image. No bound on the Galois cohomological dimension of \(k\) enters this relative calculation.

For the sheaf form of the cover construction and its cohomological comparison, see the Stacks Project, §20.24 and §59.19.

One detail in that affine-bound proof is useful here. Its boundary calculation, Theorem 5.1, Interlude A, uses a local domain \(R\) containing a separably closed field \(L\), and \(0\ne f\in\mathfrak m_R\). The element \(f\) is transcendental over \(L\): a nonzero algebraic element has its inverse in \(L[f]\), contradicting membership in the maximal ideal. Moreover \(R[1/f]\) contains the whole rational function field \(L(f)\). For every nonzero polynomial write \(P(T)=T^r(c+TQ(T))\), with \(c\ne0\). Then \(c+fQ(f)\) is a unit in the local ring, and \(P(f)\) becomes a unit after inverting \(f\). This supplies every denominator of \(L(f)\). The strict-local finite models have fraction fields of the dimension stated there, so the smaller-dimensional affine induction applies to their finitely generated \(L(f)\)-subalgebras. The support-drop proof in Theorem 6.2 then uses those affine bounds and continuity, not ordinary geometric-fibre cohomology for a nonproper map.

Let \(k_s\) be a separable closure of \(k\). The morphism \(\operatorname{Spec}k_s\to\operatorname{Spec}k\) is an affine inverse limit of finite étale morphisms. Étale localization and the continuity argument AL.4 identify pullback of \(Rf_*F\) with the corresponding direct image over \(k_s\). The extension from \(k_s\) to an algebraic closure is purely inseparable; étale topological invariance identifies the two sites and their direct-image functors. These pullbacks are conservative. Constructibility descends along this surjective quasi-compact change of base by the finite-presentation detection lemma in the extension-by-zero companion. Thus it suffices to prove ordinary finiteness over algebraically closed fields, simultaneously in all characteristics prime to \(n\).

We record the geometry needed to choose a smooth dense open, including positive characteristic. If \(L/k\) is a finitely generated field extension of transcendence degree \(d\), with \(k\) perfect of characteristic \(p>0\), choose any transcendence basis \(t_1,\ldots,t_d\) and put \(e=[L:k(t_1,\ldots,t_d)]\). Frobenius is an isomorphism of the extensions \(L/k(t)\) and \(L^p/k(t^p)\). Comparing degrees over \(k(t^p)\) gives

Choose successively \(z_i\) outside \(L^p(z_1,\ldots,z_{i-1})\) until that field is \(L\). Each step has degree \(p\), so there are exactly \(d\) steps and the monomials \(z_1^{a_1}\cdots z_d^{a_d}\), \(0\leq a_i<p\), are a basis over \(L^p\). The \(z_i\)'s are algebraically independent over \(k\). Indeed, regroup a polynomial relation of least positive total degree according to the exponents modulo \(p\). Perfection of \(k\) writes it as \(\sum_a z^a P_a(z)^p=0\). Independence of those monomials makes every \(P_a(z)\) zero, whereas each nonzero polynomial \(P_a\) has smaller degree, a contradiction.

The finite extension \(L/k(z)\) is separable. To see this without a differential criterion, let \(E\) be its maximal separable subextension. There is one \(r\) with \(L^{p^r}\subset E\): for each of finitely many algebraic generators, its minimal polynomial is a separable polynomial in some \(p\)-power of the variable. But \(L=L^p k(z)\), hence by iteration \(L=L^{p^r}k(z)\subset E\). In characteristic zero any transcendence basis is separating. A primitive element for this finite separable extension, after clearing its monic polynomial's coefficients and inverting its discriminant, gives a finite étale algebra over an open of \(\mathbf A^d_k\). The given integral affine variety and this algebra have the same fraction field; expressing their finite lists of generators as fractions and inverting the denominators identifies nonempty open subsets of the two. Hence every reduced variety over a perfect field has a smooth dense open. For several components first remove their intersections. Intersect this open with the generic locally constant loci of a constructible sheaf. The result is smooth, the sheaf is lisse there, and the complement has smaller dimension.

The dimension statements used here follow from the monic-elimination proof of Noether normalization in K.1: that proof does not use characteristic zero. It makes the dimension of an integral affine algebra equal to the transcendence degree of its fraction field. In particular, on every component on which a regular function \(t\) is nonconstant, the generic fibre over \(k(t)\) has dimension one less. A component on which \(t\) is constant has empty generic fibre. Algebraic field extension preserves these dimensions, as is also seen by base changing the finite normalization map to affine space.

M.2. A finite-support lemma for an initially arbitrary sheaf

Assume \(k\) algebraically closed. Let \(Y\) be affine, embedded as a closed subscheme of \(\mathbf A^a_k\), with coordinate functions \(t_i\). Write \(Y_{\bar\eta_i}\) for the fibre over an algebraic closure of \(k(t_i)\). Suppose a \(\Lambda\)-sheaf \(E\), not yet assumed constructible, has constructible restriction to each \(Y_{\bar\eta_i}\). Then there is a constructible subsheaf \(G\subset E\) such that \(E/G\) is supported on closed points.

Here are the finite-presentation details. On a Noetherian scheme every sheaf of finite-ring modules is the filtered union of constructible subsheaves: take the images of finite sums of the étale generators \(u_!\Lambda\); each image is constructible by the Noetherian subobject and quotient lemma. Pullback is exact and commutes with this union. The constructible sheaf \(E|Y_{\bar\eta_i}\) is finitely presented, hence compact for filtered colimits. Its identity therefore factors through one member of the pulled-back union. Since that member is a subsheaf, it equals the whole restriction. Choose one such subsheaf for every \(i\), and let \(G\) be their finite sum. The quotient has zero restriction to every coordinate geometric generic fibre.

If \(y\in Y\) is not closed, then \(\kappa(y)\) has positive transcendence degree over \(k\). Some \(t_i(y)\) is transcendental: these coordinates generate the residue field as a field. The point lies over that coordinate's generic point. After extending its geometric field to contain the chosen algebraic closure of \(k(t_i)\), its stalk is a stalk of the zero generic restriction. Thus \((E/G)_{\bar y}=0\).

This argument works on finitely many affine opens of a general \(Y\). If \(G_i\subset E|_{U_i}\) are the resulting subsheaves and \(v_i:U_i\to Y\), take the sum of the images of \(v_{i!}G_i\to E\). These images are constructible, and the resulting quotient is supported on closed points on each open, hence on \(Y\). Closed points here are the points with residue field algebraic over \(k\), so closedness agrees on the affine charts.

A sheaf \(Q\) supported on closed points is the filtered union of constructible sheaves with finite support. A constructible subset containing only closed points is finite: each of its finitely many locally closed pieces has dimension zero, since a positive-dimensional piece contains a nonclosed generic point. Such a finite-support sheaf is the finite sum of its point sheaves, by the closed-open stalk criterion. A closed point over \(k\) has a trivial étale topos, so its section functor is exact. Filtered cohomological continuity gives

The direct sum description can equally be checked on stalks of the filtered finite-support subsheaves. It does not assume that an arbitrary sheaf has finite stalks. In particular, if \(\Gamma(Y,Q)\) is finite, then \(Q\) is constructible and has finite support.

M.3. Recovering finiteness on a proper scheme

Let \(Y/k\) be proper and \(C\in D^b(Y,\Lambda)\). Suppose \(R\Gamma(Y,C)\) has finite cohomology and that, on each affine coordinate chart, every \(\mathcal H^qC\) has constructible restrictions to the coordinate geometric generic fibres. Then \(C\in D^b_c(Y,\Lambda)\).

Apply M.2 in each of the finitely many nonzero degrees. We obtain

with \(G_q\) constructible and \(Q_q\) supported on closed points. Proper finiteness from Proposition 3.2 and (M.2) show that \(H^b(Y,\mathcal H^qC)\) is finite for every \(b>0\). In the bounded hypercohomology spectral sequence

all entries with \(b>0\) are therefore finite. No differential enters column zero. Every differential leaving it has finite image, since its target is a subquotient of a finite entry with positive first index. Only finitely many differentials can occur. The final column-zero term is a subquotient of the assumed finite abutment. Working backwards through these finitely many kernels proves that \(H^0(Y,\mathcal H^qC)\) is finite. The exact sequence of (M.3) then puts \(H^0(Y,Q_q)\) between a quotient of this finite group and a subgroup of the finite group \(H^1(Y,G_q)\). It is finite. By (M.2), \(Q_q\), and therefore \(\mathcal H^qC\), is constructible. This proves the assertion.

M.4. The ordinary direct-image theorem

Theorem. For every separated finite-type morphism \(f:X\to Y\) over a field, \(Rf_*\) takes \(D^b_c(X,\Lambda)\) to \(D^b_c(Y,\Lambda)\).

By M.1, we may work over algebraically closed fields, and boundedness is already known. It is enough to treat a constructible sheaf, then use its cohomology triangles. Induct on \(\dim X\), over all algebraically closed fields simultaneously. Dimension zero follows from exact finite direct image: a zero-dimensional finite-type source is finite over the field, and its morphism to the separated target is finite, by its closed graph. Reduction and nilpotent thickenings do not change the étale site.

At dimension \(d>0\), first let \(j:V\to W\) be an open immersion, with \(V\) smooth and \(L\) finite lisse on \(V\). Discard the part of \(W\) outside the closure of \(V\), using exact closed direct image. Localize \(W\) to an affine open. Its projective closure \(\bar W\) has the same dimension; \(V\) remains open in it. It suffices to prove constructibility of \(Rj_*L\) on this proper closure.

Cover \(\bar W\) by affine coordinate charts. Over each coordinate geometric generic point the source has dimension at most \(d-1\), by M.1. The base change from the coordinate line to \(k(t_i)^{\mathrm{sep}}\) is an affine inverse limit of étale changes of base: finite separable extensions spread to finite étale covers after removing finitely many points. Étale localization and AL.4, followed by purely inseparable topological invariance, identify

Induction makes the right side bounded constructible. Its source need not be smooth over the new field; the induction statement covers all sources.

The other hypothesis of M.3 is global finiteness. Write \(V\) as the finite disjoint union of its smooth pure-dimensional components. On a component of dimension \(e\), Proposition 3.3 and the internal-Hom adjunction give

Indeed the dual of the lisse sheaf \(L^\vee(e)[2e]\) is \(L\). The finite-module argument at the start of M.6 proves self-injectivity and evaluation independently of any geometry; the finite-free resolution calculation in M.5 sheafifies it with the actual pairing. This step does not assume that \(L\) is free or has finite projective dimension, and does not use singular biduality. Compact-support finiteness proves finiteness of the right side. Also \(R\Gamma(\bar W,Rj_*L)=R\Gamma(V,L)\). Apply M.3 with (M.5). This proves the open-immersion assertion for smooth lisse source in dimension \(d\).

Next let \(f:V\to Y\) have smooth source and lisse coefficient sheaf. On an affine target cover \(V\) by finitely many affine opens. Their intersections are affine. For each such affine source, a closed embedding into affine space over the affine target and projective closure of its graph factor the map as

The open case just proved makes \(Rj_*L\) constructible; Proposition 3.2 makes \(Rp_*Rj_*L\) constructible. The finite Čech-to-derived-image sequence for the affine cover proves the assertion for the whole smooth source. This sequence is for ordinary images of the intersections; it does not replace them by extensions by zero.

Finally take any reduced \(X\) of dimension \(d\) and constructible \(F\). Choose the smooth dense open \(v:V\to X\) from M.1, with \(F|_V\) lisse, and let \(i:T\to X\) be its lower-dimensional complement. The cone

is bounded constructible by the smooth-open case already proved at this same dimension. Its restriction to \(V\) is zero. The localization triangle identifies \(C=i_*i^*C\); it is a bounded constructible complex on the smaller-dimensional \(T\). Induction makes \(Rf_*C\) constructible. The smooth-source case makes \(R(fv)_*(F|_V)\) constructible. Apply \(Rf_*\) to (M.8) to obtain the assertion for \(F\). This ordering is essential: constructibility of the cone is established before induction is applied to it. Field descent from M.1 completes the theorem. \(\square\)

M.5. Constructible duals before biduality

Put \(\omega_X=a_X^!\Lambda\) and \(D_XF=R\mathcal Hom(F,\omega_X)\). BF–BJ construct the operations and the specified smooth trace. We prove the restriction comparisons before using biduality.

Finite resolutions and restriction. A constructible sheaf \(F\) has a resolution \(P^\bullet\to F\), in degrees at most zero, whose terms are finite sums of \(u_!\Lambda\) with \(u\) affine étale of finite presentation. Indeed, the finite presentation in the constructible-sheaf companion gives a finite generating surjection. Its kernel is constructible by that companion's Noetherian subobject proof, so the same construction applies to the kernel. Repetition gives the asserted exact resolution; it need not have finite length.

For a bounded-below complex \(B\),

This is an equality of the canonical adjunction maps. On an étale test object, extension by zero, tensor-Hom adjunction and \(u_!\dashv u^*\) give the same Hom functor. To compute it, choose a bounded-below injective resolution of \(B\). Restriction to any étale object preserves injectives, since its left adjoint, extension by zero, is exact. Thus the sheaf Hom calculation on those test objects is the derived one, and its right side is the stated ordinary direct image.

The same observation computes \(R\mathcal Hom(F,B)\) by the total Hom complex from \(P^\bullet\) into a bounded-below injective resolution of \(B\). To justify replacing \(F\), restrict to any étale object. Hom into each injective is exact, and the augmented resolution is exact. If the target resolution starts in degree \(b\), its Hom double complex has indices \(r=-\deg P\geq0\) and \(s\geq b\), with total degree \(r+s\). Every diagonal is finite. The augmented exact rows therefore give the required quasi-isomorphism. Equivalently, finite truncations stabilize in each degree; no infinite product must commute with a pullback.

Let \(q\) be induced by an étale change of a parameter base, or an affine inverse limit of such changes. Pullback takes the displayed resolution to a resolution of \(q^*F\), with terms \(u'_!\Lambda\). For each term, AL.4's ordinary-image comparison gives \(q^*Ru_*u^*B\simeq Ru'_*u'^*q^*B\). These termwise comparisons are compatible with the resolution differential, since they are adjunction mates. Their resolution spectral sequences have terms \(R^su_*u^*B\) in bidegrees \((r,s)\) with \(r\geq0\), \(s\geq b\). They are bounded below and have only finitely many terms in each total degree. Exact pullback preserves the filtration and its associated cohomology sequences. Comparing its successive finite pieces, degree by degree, proves

The map here is the transpose of pulled-back evaluation, under the tensor comparison of BH. It is the map just computed on (M.9), not a separately selected isomorphism. Cohomology triangles extend the result to bounded constructible \(F\). An equivalence of étale topoi from a purely inseparable change of field gives the same comparison directly. We retain the bounded-below condition on \(B\); finite generation alone would not justify the infinite resolution against an arbitrary target.

The dualizing complex. Locally embed \(X\) as a closed subscheme \(b:X\to\mathbf A^N_k\). Transitivity and the smooth trace of BJ give

If \(u\) is the complementary open immersion, \(b_*b^!C\) is the fibre of \(C\to Ru_*u^*C\). This is the closed-support localization triangle, with its adjunction unit and counit. Define its closed-support comparison by pulling back the counit \(b_*b^!C\to C\), applying exact closed-image base change, and transposing through \(b'_*\dashv b'^!\). Naturality gives a map of the localization triangles. AL.4 identifies the pullback of the open direct-image term with its counterpart upstairs, and the middle term is exact pullback of \(C\). The cohomology sequences therefore prove that the closed-support comparison is an isomorphism. This construction specifies its counit; it does not choose functorial cones in an abstract triangulated category.

The lower bound of (M.11) follows from this fibre construction. Its upper bound follows from the relative bound on \(Ru_*\) in M.1. Both are local bounds, and a finite affine cover gives bounds on \(\omega_X\) itself. This establishes boundedness before constructibility of \(\omega_X\) or its duals is asserted.

A separable algebraic field extension is an affine inverse limit of finite étale field extensions. The preceding closed-support comparison and (M.11) therefore identify its pullback of \(\omega_X\) with the structural dualizing complex upstairs: the ambient dimension and Tate orientation are unchanged, by BJ. Purely inseparable algebraic extension is an equivalence of the relevant étale topoi and gives the same identification. Combining with (M.10), duals commute with extension to an algebraic closure. Since (M.10) is induced by evaluation and the closed-support map preserves the counit, these comparisons also preserve the dual evaluation maps. A general separable field extension would additionally use full smooth base change; it is not needed for this reduction.

Lisse coefficients without a projective-dimension assumption. The coefficient argument at the start of M.6 proves that \(\Lambda\) is injective as a module over itself and that finite-module evaluation is biduality. If \(M\) is a finite \(\Lambda\)-module, choose a resolution by finite free modules in nonpositive degrees: kernels are finite, so finitely many generators suffice at every step. On a trivializing étale open, its constant-sheaf version is exact on stalks. The same finite-diagonal calculation as above computes internal derived Hom into \(\Lambda\) by the sheafified module Hom complex. Its positive cohomology vanishes by self-injectivity, and its degree zero is the constant sheaf \(M^\vee\). Evaluation is the sheafification of module evaluation, because the comparison uses the Hom evaluation pairing throughout. Thus for a finite lisse \(L\) on a smooth pure \(e\)-dimensional scheme, \(D_VL=L^\vee(e)[2e]\), with its canonical pairing. The resolution may be infinite; no finite Tor dimension has been assumed.

We can now establish bounded constructibility of \(D_XF\). Work over an algebraically closed field, then descend by the comparison just proved and constructibility detection in M.1. Induct on dimension, choosing a smooth dense open \(v:V\to X\), with \(F|_V\) lisse, and complementary closed immersion \(i:T\to X\). Tensor-Hom adjunction, the projection formula and transitivity give

For example, testing the first identity against \(E\) replaces \(E\otimes^Lv_!L\) by \(v_!(v^*E\otimes^LL)\); transpose its map to \(\omega_X\) through \(v_!\dashv v^*\). The result is Hom from \(E\) to \(Rv_*R\mathcal Hom(L,v^*\omega_X)\), and \(v^*\omega_X=\omega_V\). For the second, use the exact proper image \(i_*\), its projection formula and \(i^!\omega_X=\omega_T\). These are the evaluation-and-counit isomorphisms. No double dual is used in their construction.

The lisse calculation and Theorem M.4 make the open term of (M.12) bounded constructible. Induction handles the smaller-dimensional closed term. Apply \(D_X\) to the triangle \(v_!F|_V\to F\to i_*i^*F\), and then use the finite cohomology filtration for a bounded constructible complex. This proves preservation of \(D^b_c\) before asserting biduality. The use of M.4 is not circular: its smooth-source step needs only the independent finite-module calculation and the smooth trace, not singular biduality.

Generic restriction and evaluation. On an affine \(X\subset\mathbf A^N_k\), let \(t\) be an ambient coordinate and let \(q:X_{\bar\eta}\to X\) be its geometric generic fibre. A finite separable extension of \(k(t)\) spreads to a finite étale cover of a nonempty open of the coordinate line: clear coefficients of a separable defining polynomial and invert its discriminant. These charts form the required affine inverse system. Purely inseparable extension then gives the algebraic closure. Thus the comparisons above apply to this \(q\).

The new ambient scheme is \(\mathbf A^{N-1}_{\overline{k(t)}}\). Pull back (M.11) using the closed-support comparison. Its twist is still \((N)[2N]\), whereas structural duality over the new field uses \((N-1)[2N-2]\). The invertible-twist comparison (BH.21) therefore gives exactly

To track double evaluation, write \(T=\Lambda(1)[2]\) on the fibre. Internal Hom sends tensor with this invertible object in its first argument to tensor with \(T^{-1}\). Consequently two applications of (M.13) identify the pulled-back double dual with \(D_{\bar\eta}D_{\bar\eta}(q^*F)\otimes T^{-1}\otimes T\), followed by the evaluation \(T^{-1}\otimes T\to\Lambda\). Under this identification the pulled-back map \(F\to D_XD_XF\) is canonical double evaluation on the fibre. Indeed the first comparison is adjoint to evaluation; apply that identity twice and the tensor-Hom triangle identities cancel the two inverse-line maps. The shift in \(T\) is even, so moving these line factors creates no extra sign. This checks the actual map needed in the dimension induction, not merely its source and target objects.

M.6. Singular constructible biduality

We first supply the coefficient calculation, including its extension step. Put \(\Lambda=\mathbf Z/n\). Suppose \(A\subset B\) are \(\Lambda\)-modules and \(h:A\to\Lambda\) is linear. For \(b\in B\), the scalars \(r\) with \(rb\in A\) form an ideal \(I=d\Lambda\), with \(d\mid n\). Indeed the inverse image of \(I\) in \(\mathbf Z\) is an ideal containing \(n\mathbf Z\); division by its least positive element gives this description. Write \(a=h(db)\). The relation \((n/d)db=0\) implies \((n/d)a=0\), so \(a=dc\) in \(\Lambda\) for some \(c\). Define \[ \widetilde h(x+rb)=h(x)+rc\qquad(x\in A,\ r\in\Lambda). \] This is well defined: if \(tb\in A\), write \(t=ds\); then \(h(tb)=sa=tc\). Thus every partial extension extends across one more generator. Order partial extensions by inclusion; their union along a chain is a partial extension. The maximality principle therefore gives an extension to all of \(B\). This proves that \(\Lambda\) is injective and that \(M\mapsto M^\vee=\operatorname{Hom}_\Lambda(M,\Lambda)\) is exact.

For a finite module, duality also preserves cardinality and separates points. A cyclic subgroup \(C\) of order \(d\mid n\) has exactly \(d\) homomorphisms to \(\Lambda\), since the image of its generator is an element annihilated by \(d\). Sending that generator to \(n/d\) embeds \(C\) in \(\Lambda\); the extension just proved supplies a functional on the entire module which is nonzero at any specified nonzero element of \(C\). For \(0\ne C\subset M\), exactness gives \[ 0\longrightarrow(M/C)^\vee\longrightarrow M^\vee \longrightarrow C^\vee\longrightarrow0. \] Induction on \(|M|\) yields \(|M^\vee|=|M|\), including the zero module as the starting case. Evaluation \(M\to M^{\vee\vee}\) is injective by separation and surjective by this equality applied twice.

Exactness of duality applied to the cycle and boundary sequences of any complex gives \[ H^i\!\left(R\operatorname{Hom}_\Lambda(A,\Lambda)\right) =\operatorname{Hom}_\Lambda(H^{-i}(A),\Lambda). \] Here ordinary Hom into the injective module computes derived Hom: it sends acyclic complexes to acyclic complexes by those same sequences. Therefore canonical double evaluation is an isomorphism for every complex with finite cohomology modules. In particular it applies to a bounded complex with finite cohomology; neither a free resolution of finite length nor a classification of finite abelian groups is needed.

Theorem. For every \(K\in D^b_c(X,\mathbf Z/n)\), the canonical evaluation

is an isomorphism, without a finite-Tor-dimension assumption on \(K\).

Over an algebraically closed field, first consider proper \(X\). The internal-Hom adjunction and its trace identify

We specify both this isomorphism and its evaluation compatibility. For any \(V\in D(\Lambda)\), tensor-Hom adjunction, \(Ra_{X!}\dashv a_X^!\) and the projection formula identify \[ \begin{aligned} \operatorname{Hom}(V,R\Gamma D_XK) &=\operatorname{Hom}(a_X^*V\otimes^LK,a_X^!\Lambda)\\ &=\operatorname{Hom}(V\otimes^LR\Gamma_c K,\Lambda). \end{aligned} \tag{M.15a} \] Yoneda gives (M.15), because compact and ordinary supports agree on this proper scheme. Write its map as \(\Phi_K:T(D_XK)\to(TK)^\vee\), where \(T=R\Gamma(X,-)\) and \(V^\vee=R\operatorname{Hom}_\Lambda(V,\Lambda)\). Its pairing is cup product, evaluation into \(\omega_X\), and the trace \(T\omega_X\to\Lambda\), in that order.

Let \(\eta\) denote canonical double evaluation. Then \(\Phi_{D_XK}\,T(\eta_K)=\Phi_K^\vee\,\eta_{TK}\). To check this equality, transpose both sides to pairings with \(T(D_XK)\). The defining identity for \(\eta_K\) says that evaluation of \(D_XK\) against \(\eta_K(K)\) is evaluation of \(K\) against \(D_XK\), after the tensor symmetry. Naturality and symmetry of cup product give the same identity after \(T\); composing with the same trace preserves it. The module evaluation on the right uses that same tensor symmetry, so the Koszul sign agrees. This proves equality of the maps in the derived category.

Proper finiteness makes \(TK\) bounded with finite cohomology. The coefficient calculation above makes \(\eta_{TK}\) an isomorphism, and both \(\Phi\)'s are the adjunction isomorphisms just proved. Therefore \(T(\eta_K)\) is an isomorphism. Thus the cone \(\Delta_K\) of (M.14) satisfies

Induct on \(\dim X\), simultaneously over all algebraically closed fields. Dimension zero is finite-module biduality on finitely many points. Suppose the assertion proved below dimension \(d\). On every affine coordinate chart of a proper \(d\)-dimensional \(X\), the coordinate geometric generic fibres have dimension at most \(d-1\). By (M.13) and induction, \(\Delta_K\) restricts to zero on each such fibre. M.5 has already proved that its cohomology sheaves are bounded constructible. The coordinate argument of M.2 shows that they are supported on closed points, and constructibility makes that support finite. Formula (M.2) now identifies

Each finite-support sheaf is detected by its global sections. Equations (M.16)–(M.17) force every \(\mathcal H^r\Delta_K\) to be zero. This proves biduality on all proper schemes of dimension \(d\).

For a general \(X\), the assertion is local on affine opens: restriction commutes with duality by (M.10)–(M.11). Embed an affine open into its projective closure of the same dimension and extend \(K\) by zero. That extension is bounded constructible; the proper case just proved applies to it. Restrict its evaluation isomorphism back to the open. This proves the induction step for every \(X\). Finally the field-extension comparisons of M.5 and conservativity descend the canonical isomorphism to arbitrary \(k\). \(\square\)

The finite-support step is why vanishing of global sections suffices here. It would be false to infer an arbitrary constructible complex is zero merely from vanishing of its global cohomology.

M.7. Exchanges, coefficients and the remaining foundations

The internal-Hom adjunction and transitivity give, before biduality,

Here are the two tests, which also specify the maps. Test the first identity against \(E\in D(Y,\Lambda)\). Tensor-Hom adjunction turns its left side into \(\operatorname{Hom}(E\otimes^LRf_!K,\omega_Y)\). The projection formula and the exceptional adjunction turn this into \(\operatorname{Hom}(f^*E\otimes^LK,f^!\omega_Y)\). Transitivity identifies \(f^!\omega_Y=\omega_X\); tensor-Hom and \(f^*\dashv Rf_*\) give \(\operatorname{Hom}(E,Rf_*D_XK)\). For the second identity, test against \(E\in D(X,\Lambda)\) and use \(\operatorname{Hom}(E,f^!D_YA)=\operatorname{Hom}(Rf_!E\otimes^LA,\omega_Y)=\operatorname{Hom}(E,D_Xf^*A)\). Every equality is natural in the test object, so Yoneda proves (M.18). Each chain transposes evaluation through the projection formula and the composed counit. It consequently identifies the same double-evaluation maps used in M.6, rather than only producing isomorphic objects. M.4–M.6 and Proposition 3.2 show that all the objects needed to apply biduality are bounded constructible. Replace \(K\) by \(D_XK\) in the first identity, then dualize, to obtain

In particular \(f^!\) preserves bounded constructibility. For a closed geometric point over an algebraically closed field, the closed-immersion form of (M.18)–(M.19) proves both point exchanges in (2.1), with finite-module dual over \(\mathbf Z/n\). Over an arbitrary field they are the corresponding geometric closed-point statements, by separable extension and topological invariance. These untwisted point formulas concern closed points; generic restriction instead has the dimension correction explicitly displayed in (M.13).

The same arguments prove M.5–M.6 over every finite coefficient field: finite-dimensional linear duality replaces the self-injective \(\mathbf Z/n\)-module calculation; all geometric finiteness and comparison inputs hold for its underlying finite torsion sheaves. With these coefficients, tensor and internal Hom also preserve bounded constructibility. For tensor, use exact tensor on stalks and a common constructible partition; bounded complexes have a finite cohomology filtration. For internal Hom use the adjunction identity

proved by tensor-Hom adjunction and (M.14). The tensor on the right is now bounded constructible, so its dual is too. Over \(\mathbf Z/\ell^r\) with \(r>1\), this bounded tensor/internal-Hom conclusion requires a finite-Tor-dimension hypothesis on the first argument. It is not needed for (M.14): a finite module can have infinite projective dimension while its dual against the self-injective ring is exact. This distinguishes the two coefficient claims.

Thus ordinary finite-coefficient direct-image finiteness, preservation by structural duality, singular biduality and the dual exchanges over fields have written arguments here. They do not establish the normalized adic operations or their rationalization comparisons. The inputs of Propositions 3.2–3.3 include compactification and curve completion, curve finiteness and duality, étale cohomological continuity and dimension bounds, and the categorical exceptional adjoint. This field argument does not imply generic local acyclicity over a trait or absolute purity for arbitrary regular schemes.

Appendix N. Finite monodromy, compatible lattices and congruence covers

The arithmetic arguments use two different kinds of finiteness. A finite local system is trivialized by one finite étale cover. A lisse adic system generally is not: each finite reduction has such a cover, but the covers may grow with the reduction level. We construct the relevant monodromy equivalences and then obtain the single congruence cover needed to remove wild prime-to-\(\ell\) monodromy.

Throughout this appendix \(X\) is connected and Noetherian, with a geometric point \(\bar x\). In its application to this lesson, \(X\) is a connected stratum of a finite-type scheme over a field. All finite covers below are étale. We use the elementary affine description of such covers as finite locally free algebras with vanishing relative differentials. The effective descent and fundamental-group equivalence used below are proved here, rather than imported as citations to the corresponding general theorems.

N.1. Descend a finite algebra before naming its monodromy

First we give the affine descent calculation. Let \(A\to B\) be faithfully flat and let \(M\) be a \(B\)-module with a descent isomorphism \[ \theta:M\otimes_A B\longrightarrow B\otimes_A M \tag{N.1} \] over \(B\otimes_A B\), satisfying the cocycle identity over three factors. Put \(\delta(m)=\theta(m\otimes1)\). If \(\delta(m)=\sum b_i\otimes m_i\), the cocycle and diagonal identities say \[ \sum b_i\otimes\delta(m_i)=\sum b_i\otimes1\otimes m_i, \qquad \sum b_i m_i=m. \tag{N.2} \] Indeed the two sides of the first identity apply the overlap isomorphism twice or once to \(m\otimes1\otimes1\); pulling the cocycle to the diagonal makes its invertible restriction idempotent, hence the identity. Also \(\delta(bm)=(b\otimes1)\delta(m)\).

Let \(N=\{m\in M:\delta(m)=1\otimes m\}\). Flatness identifies \(B\otimes_A N\) with the equalizer of the two maps from \(B\otimes_A M\) to \(B\otimes_A B\otimes_A M\) in (N.2). Thus \(\delta\) takes values in \(B\otimes_A N\). It is inverse to multiplication \[ B\otimes_A N\longrightarrow M,\qquad b\otimes n\longmapsto bn. \tag{N.3} \] One composition is the second identity of (N.2); the other uses \(\delta(bn)=b\otimes n\). A map respecting descent data preserves these equalizers, so the construction also descends every compatible module map, uniquely.

For completeness, starting with an \(A\)-module \(Q\) recovers exactly \(Q\) from \(B\otimes_A Q\). The equalizer sequence \[ 0\longrightarrow Q\longrightarrow B\otimes_A Q \rightrightarrows B\otimes_A B\otimes_A Q \tag{N.4} \] is exact: after tensoring with \(B\), multiplication of the added factor with the first displayed factor contracts the augmented alternating complex. Inserting a unit in the first position gives the identity; all other insertions cancel in adjacent pairs. Flat tensor preserves its cohomology, and faithful flatness detects zero cohomology. This proves (N.4), including injectivity, before descent is used.

If \(M\) is a \(B\)-algebra and \(\theta\) respects its multiplication and unit, then \(N\) is an \(A\)-algebra and (N.3) is an algebra isomorphism. Suppose \(A\) is Noetherian and \(M\) is finite locally free over \(B\). Express a finite set of \(B\)-generators through (N.3). The finitely many elements of \(N\) occurring in those expressions generate \(N\): their cokernel vanishes after the faithfully flat base change. Thus \(N\) is finite and finitely presented over \(A\). It is flat, since injectivity after tensoring any injection with \(N\) can be tested after \(B\), where it is flatness of \(M\). It is therefore locally free. Explicitly, over a local ring choose lifts of a residue-field basis of \(N\). Nakayama gives a surjection from a finite free module; flatness makes its kernel zero modulo the maximal ideal, and finite generation plus Nakayama makes the kernel zero. This also proves the asserted local freeness.

If \(M\) is finite étale, then \(\Omega_{N/A}\otimes_A B=\Omega_{M/B}=0\); the equality follows from the universal property of differentials. Faithfulness gives \(\Omega_{N/A}=0\), so the descended finite locally free algebra is étale. The construction commutes with localizations, since flat localization preserves the defining equalizer. It therefore glues over affine opens, and its uniqueness makes the overlap identifications and all cocycles agree.

Now let \(\mathcal F\) be a finite locally constant set sheaf on \(X_{\mathrm{ét}}\). On an affine base open choose finitely many affine members of a trivializing étale cover; this is possible because étale maps are open and the base is quasi-compact. Their disjoint union is a faithfully flat affine cover. The finite constant sheaves there are represented by disjoint unions of copies of the cover. Their transition identifications give descent data on the finite product algebras, and the calculation above descends those algebras and their maps. Gluing the affine answers proves that \(\mathcal F\) is represented by a finite étale \(X\)-scheme. Conversely, the ordering cover of a finite étale scheme trivializes its represented sheaf, as constructed next. The same calculation on maps proves an equivalence, not merely existence of a representative.

N.2. Pointed ordering covers supply the finite quotients

Let \(Y\to X\) be a nonempty finite étale cover, of constant degree \(d\). Inside \(Y^d_X\), remove every pairwise diagonal. Each diagonal is open because the map is étale, and closed because it is finite and separated. The resulting ordering cover \(P\to X\) is finite étale; its geometric fibres are the orderings of a \(d\)-element set. Its tautological sections split \(Y_P\) into \(d\) copies of \(P\). A morphism between finite étale covers which is bijective on all geometric fibres is an isomorphism: on affine bases this is the map between finite locally free modules, checked after each residue field and then by the local basis criterion.

Choose an ordering above \(\bar x\), and take its connected component \(P_0\). Noetherianity makes the finitely many connected components open and closed. Each component of a nonempty finite étale cover maps onto the connected base: its image is both open and closed. The permutation group on \(d\) letters acts simply transitively on every fibre of \(P\). The stabilizer of \(P_0\) therefore acts simply transitively on every geometric fibre of \(P_0\). Indeed, the unique permutation carrying two fibre points to each other carries their connected components to each other. It follows that \[ \coprod_{a\in G_P}P_0\longrightarrow P_0\times_X P_0, \qquad z\longmapsto(z,a(z)) \tag{N.5} \] is an isomorphism, where \(G_P=\operatorname{Aut}_X(P_0)\). There are no additional automorphisms: two maps from a connected cover to an étale separated cover agreeing at one geometric point agree everywhere. Their equalizer is open and closed. Thus an automorphism is determined by its value at the chosen point.

Call a connected pointed cover with (N.5) a pointed Galois cover. Every pointed finite cover is dominated by one: use its ordering cover, with the specified point in the first position. Two pointed Galois covers have a common such refinement by taking the component containing the chosen pair in their fibre product. The product of their deck groups acts on that product; the stabilizer of the component is again simply transitive on its fibres, proving (N.5) for the refinement. Pointed maps are unique by the equalizer argument.

For a pointed map \(r:Q\to P\), every \(b\in G_Q\) determines a unique \(a\in G_P\) with \(rb=ar\): choose \(a\) to give the same value at the distinguished point, and then use the same equalizer argument. This defines a surjective group homomorphism \(G_Q\to G_P\). Surjectivity follows because \(r\) is a nonempty finite étale map to a connected cover, so is onto on every geometric fibre, and \(G_Q\) acts transitively there.

Define the profinite group \[ \Pi=\varprojlim_{(P,\bar p)}G_P. \tag{N.6} \] Every projection to \(G_P\) is onto. To lift a prescribed element, any finite collection of compatibility conditions can be solved in a common dominating cover, using the surjective homomorphisms just proved. The corresponding closed subsets of the product of the finite groups have the finite-intersection property, so compactness gives a compatible family. The same argument gives compactness of the inverse limit. Kernels of these projections form a neighbourhood basis at the identity; common refinements account for finite intersections.

N.3. The fibre functor and all of its morphisms

If \(P\) trivializes \(Y\), evaluation at its distinguished point gives a bijection \[ \operatorname{Hom}_X(P,Y)\longrightarrow Y_{\bar x}. \tag{N.7} \] A map is a section of \(Y_P\), and on connected \(P\) such a section chooses exactly one of its constant sheets. Give the left side the left \(G_P\)-action \(a\cdot h=h\circ a^{-1}\). The inverse is essential: it makes \(a\cdot(b\cdot h)=(ab)\cdot h\). Refinements respect this action, because \(rb=ar\). Equation (N.7) thus gives a well-defined continuous \(\Pi\)-action on every finite fibre, natural in \(Y\).

Conversely, a finite continuous \(\Pi\)-set \(S\) factors through some \(G_P\). The kernel of its finite permutation action is open and therefore contains one projection kernel; surjectivity of the projection makes the factorization well-defined. On \(P\times S\), let a deck transformation act by \[ a(z,s)=(a(z),a\cdot s). \tag{N.8} \] The torsor identity (N.5) turns this action into a descent datum on the disjoint union of copies of \(P\). N.1 descends it to a finite étale \(X\)-scheme. At the distinguished point, its fibre is \(S\), with the prescribed action: the section labelled \(s\), evaluated at \(a^{-1}\bar p\), is identified by (N.8) with the label \(a\cdot s\) at \(\bar p\).

Two covers are split by a common pointed Galois cover. A map of their fibres is \(\Pi\)-equivariant exactly when it is equivariant for that finite deck group. Such a map between the constant sheets respects their descent data, so N.1 gives a unique map of covers. We have proved full faithfulness and essential surjectivity of the fibre functor.

Finally \(\Pi\) is the automorphism group of that functor, not merely a group producing an equivalent category. A natural automorphism of the forgetful functor on finite \(\Pi\)-sets is determined on every regular quotient \(G_P\). It commutes with the equivariant right translations, so is left translation by a unique element of \(G_P\). Naturality for quotient maps makes these elements compatible. Conversely, their compatible family acts naturally on every finite set. The topology also agrees, since finite fibres define both neighbourhood bases. Thus \(\Pi=\pi_1(X,\bar x)\).

Finite products are preserved by fibre evaluation. Consequently group laws, module operations over a finite ring and all their identities descend through this equivalence. This proves the finite locally constant module/continuous finite-module correspondence, including morphisms, tensor products and quotient modules. Pullback is compatible as well: pulling back finite covers commutes with their geometric fibres and hence induces the corresponding continuous homomorphism of fibre-automorphism groups.

N.4. Recover the integral module from every finite level

Let \(\mathcal O\) be a complete discrete valuation ring with uniformizer \(\varpi\) and finite residue field \(\kappa\). This includes the ring of integers of every finite extension of \(\mathbb Q_\ell\), as well as complete discrete valuation rings of equal characteristic. Each quotient \(\mathcal O/\varpi^n\) is finite: its power filtration has \(n\) successive quotients isomorphic to \(\kappa\), via multiplication by \(\varpi^i\) in the \(i\)-th layer. A strict lisse \(\mathcal O\)-system consists of finite locally constant \(\mathcal O/\varpi^n\)-module sheaves \(\mathcal F_n\) whose transition maps identify the quotient of level \(n+1\) by \(\varpi^n\) with level \(n\). Set \(M_n=(\mathcal F_n)_{\bar x}\).

The transition maps are onto. In \(M=\varprojlim M_n\), projection to each \(M_n\) is onto by successive lifts. Its kernel equals \(\varpi^nM\). For the nontrivial inclusion, a compatible element vanishing at level \(n\) is divisible by \(\varpi^n\) at every later level. The finite sets of possible divisors have compatible choices: any finite collection of conditions is solved by choosing a divisor at the largest index and reducing. Compactness of these finite sets gives a compatible divisor. Therefore \[ M/\varpi^nM=M_n. \tag{N.9} \] Lift a finite generating set of \(M_1\) to \(M\). Successively approximate an arbitrary element modulo \(\varpi\), divide the error and repeat. Completeness makes the resulting coefficients converge in \(\mathcal O\), so the lifts generate \(M\). Alternatively their span is dense and is the compact, hence closed, image of a finite power of \(\mathcal O\). Thus \(M\) is finite, and (N.9) identifies its adic and inverse-limit topologies.

Here are the module facts needed for the converse. Every submodule \(A\subset\mathcal O^r\) is finite free. Induct on \(r\), starting with zero. Project onto the first coordinate. If its image is zero, apply induction inside \(\mathcal O^{r-1}\). Otherwise that image contains an element of least valuation \(a\), so it is \(\varpi^a\mathcal O\). Choose \(v\in A\) whose first coordinate is \(\varpi^a\), multiplying by a unit if necessary. Subtracting the appropriate multiple of \(v\) from any element gives \[ A=\mathcal O v\oplus\bigl(A\cap(0\oplus\mathcal O^{r-1})\bigr). \] Induction proves the assertion. A finite torsion-free module embeds in its finite-dimensional fraction space: a vector killed by localization is annihilated by a nonzero scalar. Choose a basis of that space and clear the denominators of finitely many module generators. This embeds the module in a finite free module, so the same argument proves it free.

Every finite \(\mathcal O\)-module \(T\), including one with torsion, is complete and separated. To see this without assuming freeness of \(T\), present it as \(\mathcal O^r/A\). The submodule calculation just proved gives finitely many generators of \(A\). Thus \(A\) is the image of a continuous map from some \(\mathcal O^s\). Since \(\mathcal O\) is complete with finite residue field, it is the compact Hausdorff inverse limit of its finite quotients. Consequently \(A\) is compact and closed, and \(T\) is compact Hausdorff. The images \(\varpi^nT\) are exactly a neighbourhood basis of zero for this quotient topology. Their intersection is zero, and any compatible family of cosets has a common point by compactness. Hence \(T\to\varprojlim T/\varpi^nT\) is an isomorphism of topological modules.

N.3 gives compatible continuous group actions on all \(M_n\); they define a continuous action on \(M\). Conversely a continuous action on a finite \(\mathcal O\)-module gives actions on its finite quotients, and N.3 constructs a strict lisse system. The preceding completeness argument recovers the original module. Every \(\mathcal O\)-linear map of finite modules is continuous, since it preserves their \(\varpi\)-adic filtrations. It is equivariant exactly when all its reductions are. Full faithfulness at every finite level and (N.9) prove that these constructions are inverse on objects and morphisms. In particular, systems whose inverse-limit stalk is torsion free correspond to finite free \(\mathcal O\)-modules.

N.5. Rational coefficients and the one congruence cover

Let \(V\) be a finite-dimensional continuous \(E\)-representation of the profinite group \(\pi_1(X,\bar x)\). Its image is compact. Choose a lattice \(L_0\subset V\). The matrix entries of a compact set are bounded, so all translates of \(L_0\) lie in \(\varpi^{-c}L_0\) for one integer \(c\). Their \(\mathcal O\)-span \(L\) contains \(L_0\), lies in that finite free module, and is invariant under the group. The coordinate submodule calculation in N.4 proves that \(L\) is finite free; since it contains \(L_0\), it spans \(V\). It is therefore a stable lattice, and N.4 constructs its strict lisse system.

Any two lattices in \(V\) lie between bounded powers of each other: write either basis in the other basis and bound the finitely many denominators. Likewise any equivariant \(E\)-linear map becomes an equivariant integral map after multiplication by a sufficiently large power of \(\varpi\). It follows that rationalization is fully faithful and independent of the lattice: \[ \operatorname{Hom}_{\mathrm{lisse},E}(L_E,L'_E) =\operatorname{Hom}_{\pi_1,\mathcal O}(L,L')\otimes_{\mathcal O}E. \tag{N.10} \] Consequently strict lisse systems, after inverting \(\varpi\), classify exactly finite-dimensional continuous \(E\)-representations. Kernels and cokernels of their maps are continuous representations, and their invariant subspaces have stable lattices by the same argument. This is the global-lattice coefficient category. A \(\overline{\mathbb Q}_\ell\)-object here is obtained by scalar extension from some finite \(E\), as in Section 3; no assertion about arbitrary representations over an infinite coefficient extension is needed.

Reduce the stable lattice representation modulo \(\varpi\). Its kernel is an open normal subgroup of the fundamental group and hence corresponds, by N.3, to a connected finite étale cover \(p:Y\to X\). More explicitly the finite image gives the regular transitive quotient set, whose descended cover is connected; a decomposition of a finite cover would be an invariant partition into the fibres of its open and closed components.

The diagonal section of the pulled-back cover shows that \(\pi_1(Y)\) fixes the chosen point of this regular finite quotient. Its stabilizer is the reduction kernel, so the reduced representation on \(Y\) is trivial. Thus on \(Y\) the image lies in \(1+\varpi M_r(\mathcal O)\). This is a pro-\(\ell\) group. Its successive congruence quotients inject into the additive group \(M_r(\mathcal O/\varpi)\), since \[ (1+\varpi^n A)(1+\varpi^n B) \equiv1+\varpi^n(A+B)\pmod{\varpi^{n+1}} \quad(n\ge1). \tag{N.11} \] Each quotient is therefore an \(\ell\)-group, and completeness recovers the congruence group as their inverse limit. A continuous homomorphism from a pro-\(p\) group with \(p\ne\ell\) into it is trivial: its image in every finite congruence quotient is both a \(p\)-group and an \(\ell\)-group. In particular, whenever a boundary's wild inertia has been proved to be pro-\(p\), this one finite cover makes its action trivial. That ramification theorem is a separate geometric prerequisite. The conclusion is not that the entire lisse system becomes constant on the cover; its remaining pro-\(\ell\) image can be infinite.

The construction supplies finite monodromy and strict lisse coefficient foundations. It does not identify the strict rational category with all locally free rational sheaves on the pro-étale site of a singular scheme: the latter may admit lattices only locally. Nor does it prove that arbitrary constructible derived operations preserve the required integral models, commute with normalization, or have the asserted rationalization comparisons. The distinction between strict lisse lattices, constructible integral models and derived rational sheaves is treated in BW–BX. The arguments here use finite étale algebra descent, module descent and geometric-stalk comparison as specified in N.1–N.4.

Appendix O. Complete the generic smooth curve and extend its finite diagram

We prove the precise geometric assertion needed in (3.3). Let \(S\) be a finite-type scheme over a field and \(X\to S\) a smooth affine morphism of relative dimension one. For each generic point \(\eta\) of \(S\), there are an open neighbourhood \(U\), a finite surjective radicial map \(V\to U\), and \[ X_V=\coprod_iY_i\ \subset\ \coprod_iC_i, \qquad C_i\longrightarrow T_i\longrightarrow V, \tag{O.1} \] where \(T_i\to V\) is finite étale, each \(C_i\to T_i\) is smooth proper with geometrically connected one-dimensional fibres, and the complement of \(X_V\) in \(\coprod_iC_i\) is finite over \(V\). Consequently \(Y_i\) is dense in every fibre of \(C_i\to T_i\). An empty generic fibre becomes an empty family after shrinking by the direct localization calculation in AH.3.

AG.1–AG.3 prove finite normalization over every field, including inseparable extensions, with the polynomial normalization of Y.5 and the trace calculation of L.3. L.1 supplies the general normal-local-ring-to-DVR proof; AG.4 proves smoothness at a regular curve point with finite separable residue field. For finite diagrams we use the object, map, equality, open-set and finiteness proofs in Limits of schemes and Noetherian approximation, §§1–4. AH.4 verifies their finite-presentation hypotheses for these particular systems and identifies the exact proofs of the opens, sections, affineness, diagrams and finiteness assertions. O.2 supplies the required smoothness descent by finite standard-smooth charts.

O.1. A proper smooth completion after a finite purely inseparable extension

First let \(K\) be a field and \(X/K\) a smooth affine curve. Choose a closed embedding in affine space and its reduced closure \(Z\) in projective space. AH.2 checks that every component of the closure is a curve and that its boundary is finite. Its intersection with the original affine chart is exactly \(X\), and \(X\) meets every irreducible component densely. AE.2 proves that this closed projective scheme is proper by the graded closed-image calculation.

In characteristic \(p>0\), let \(F=K^{\mathrm{perf}}\), the union in an algebraic closure of all finite purely inseparable extensions of \(K\); in characteristic zero set \(F=K\). The field \(F\) is perfect: in positive characteristic the \(p\)-th root of any of its elements lies in another finite purely inseparable extension of \(K\). AH.1 identifies the underlying space of \(Z_F\) with that of \(Z\), preserving the density of \(X_F\) in every component. Normalize \((Z_F)_{\mathrm{red}}\). AG.1–AG.3 construct this normalization and prove that the resulting map \(C_F\to Z_F\) is finite. AF.1 makes that map proper, so \(C_F\) is proper. Above \(X_F\) it is an isomorphism: Y.6 makes the smooth curve regular, L.1 makes its one-dimensional local rings normal DVRs, and its generic local rings are fields. The localization of integral closure proved in AG.3 therefore leaves this normal open unchanged.

AG.3 makes the normal local rings at closed points of \(C_F\) one-dimensional, and L.1 makes them DVRs. Their residue fields are finite by Y.5 and separable because \(F\) is perfect. AG.4 constructs a smooth chart at each of these points by the split residue-field derivation calculation and the local Jacobian presentation. Its final closed-point argument shows that these charts cover \(C_F\). Thus \(C_F\) is a smooth proper curve containing \(X_F\).

All this data descends to a finite purely inseparable \(K'/K\). Here are the properties that must descend, including smoothness. The finite-presentation theorem first descends \(C_F\to Z_F\) and then makes that map finite. Descend the open \(X_F\subset C_F\), the isomorphism from it to the given \(X_F\), its inverse and both identity equations. The resulting \(C_{K'}\) is finite over the fixed projective scheme \(Z_{K'}\), and is therefore proper. Use the finite standard-smooth chart argument in O.2 to make \(C_{K'}\) smooth of relative dimension one. Every coefficient used belongs to one finite purely inseparable extension, so this process ends at such a \(K'\). In characteristic zero no extension is needed. AH.2 proves that the boundary of \(C_F\) is finite by pulling back the finite boundary of \(Z\) along the finite normalization map. The homeomorphism for \(F/K'\) then makes the reduced boundary of \(C_{K'}\) zero-dimensional, and the finite-algebra calculation there proves it finite over \(K'\).

O.2. Why the smooth charts and their dimensions descend

Let \(R=\varinjlim R_\alpha\), and suppose a finitely presented \(R\)-scheme \(D\) is smooth of relative dimension one. Cover it by finitely many affine standard-smooth charts. Such a chart has a presentation \[ \left(R[z_1,\ldots,z_m]/(g_1,\ldots,g_c)\right)_h, \qquad m-c=1, \tag{O.2} \] with a chosen \(c\)-column Jacobian minor invertible. Record a polynomial expression for its inverse in the localized quotient. Coefficients of the \(g_j\), of \(h\), and of that expression form a finite list. The equation saying that the product is one holds at a sufficiently late stage. That same presentation is consequently standard smooth there, with the same relative dimension: the conormal calculation eliminates the selected \(c\) differentials and leaves \(m-c\) free differentials. The existence of the charts, their infinitesimal lifting and differential calculations are proved in Formally smooth, unramified and étale ring maps, Theorems 4.1–5.1.

Descend each chart as an open of the already chosen model, using finite open descent and eventual affineness. Descend its isomorphism to (O.2), the inverse and the two identities. Finite open descent also makes these finitely many charts cover the model eventually. This proves the required smoothness and dimension assertions. It uses no assertion that an arbitrary geometric property descends. We apply it first to the system of finite subextensions of \(F/K\), and then to the localizations of an integral base ring.

O.3. Extend the completion over a neighbourhood of the generic point

Return to \(S\) and \(\eta\). Remove the other irreducible components and pass to the reduction. The resulting affine base \(\operatorname{Spec}A\) is integral, with fraction field \(K=\kappa(\eta)\). The reduction map to the original base is finite, surjective and radicial; it will be included in \(V\to U\). Base change preserves smoothness and affineness of \(X\).

Choose an affine embedding of \(X\) over \(\operatorname{Spec}A\), and apply O.1 to \(X_K\) using the generic fibre of this same embedding. In positive characteristic write its finite purely inseparable extension as \(K'=K(\beta_1,\ldots,\beta_r)\), with one power \(q=p^e\) such that every \(\beta_j^q\) lies in \(K\). Localize \(A\) so these finitely many elements lie in \(A\), and put \(B=A[\beta_1,\ldots,\beta_r]\subset K'\). Monomials with each exponent less than \(q\) generate \(B\) over \(A\), so it is finite; its fraction field is \(K'\). Every \(b\in B\) satisfies \(b^q\in A\). Lying over makes \(\operatorname{Spec}B\to\operatorname{Spec}A\) surjective. After extension to any algebraically closed residue field each \(\beta_j\) has a unique possible image, because \(T^q-a\) has one root. Thus every geometric fibre has one point and purely inseparable residue extension, proving radiciality. In characteristic zero take \(B=A\).

Let \(Z_B\) be the projective closure of the base-changed embedding of \(X_B\). AH.3 proves, using the homogeneous ideal and its saturation, that its generic fibre is exactly the closure used in O.1. Descend the finite map \(C_{K'}\to Z_{K'}\) to a finite map \(C\to Z_B\) after localizing \(B\). Descend the specified open, its isomorphism to \(X_{K'}\) and the inverse identities; this identifies the resulting open with all of \(X_B\). The finite map makes \(C\) proper over \(B\). Apply O.2 to make \(C\to\operatorname{Spec}B\) smooth of relative dimension one.

Choose the reduced closed structure on \(D=C\setminus X_B\). It is of finite presentation, and its generic fibre is finite by O.1. AH.4 checks the hypotheses of the finiteness proof in Theorem 4.2 of the limit lesson, which makes \(D\to\operatorname{Spec}B\) finite after another localization. In concrete terms, after making the source affine, finitely many generic module generators and the equations expressing their products by algebra generators descend; they then generate the whole algebra. This supplies finiteness of the boundary, not just its generic dimension.

All these base changes can be made over a smaller open of the original base while retaining a finite radicial map. For a localization \(B_b\), invert \(b^q\in A\): since \(b^{-1}=b^{q-1}/b^q\), one has \(B_b=B\otimes_A A_{b^q}\). In characteristic zero the same assertion uses \(q=1\). Finitely many such localizations can be combined. Write the final map as \(V\to U\).

O.4. Separate the constants and verify every fibre

Write \(V=\operatorname{Spec}B\) and \(h:C\to V\). The finite map to the projective closure constructed in O.3 makes every \(H^i(C,\mathcal O_C)\) finite over \(B\), by AF.1 and W.4. AF.2 permits a further shrinking, still of the form allowed in O.3, after which these cohomology groups commute with every change of base. In particular \(E=\Gamma(C,\mathcal O_C)\) is a finite \(B\)-algebra and \[ E\otimes_BK'=\Gamma(C_{K'},\mathcal O_{C_{K'}}). \tag{O.3} \] This is a finite product of finite separable field extensions of \(K'\). Indeed it is a finite-dimensional reduced algebra, since \(C_{K'}\) is smooth. The same equalizer commutes with field extension; smoothness makes its scalar extension to an algebraic closure reduced too. A finite field factor containing an inseparable element would contain a polynomial quotient with a repeated root after that extension, and hence a nonzero nilpotent. Tensoring the inclusion of the subfield generated by that element remains injective, a contradiction.

After shrinking \(V\), the algebra \(E\) itself is finite étale. Write each field factor of (O.3) as a finite tower of simple separable extensions. Descend the monic minimal polynomials and the Bezout equations that invert their derivatives, clearing their finite lists of denominators. They define a finite free étale tower over a localization of \(B\), by the square-zero root correction in AB.1. Let \(D\) be the product of these towers. The generic algebra isomorphism \(D\otimes_BK'\simeq E\otimes_BK'\) descends explicitly: \(D\) is finite free, and AF.2 has made \(E\) finite free, so write it and its inverse as matrices in those bases. Clear their denominators and the finitely many equations asserting mutual inverse, preservation of the unit, and preservation of products of basis elements. These matrices then define an algebra isomorphism \(D\simeq E\) over that localization. Thus \(E\) is finite étale. The \(b^q\) argument of O.3 makes this an additional shrinking of \(U\), preserving the finite radicial map \(V\to U\) and the base-change conclusion of AF.2.

Put \(T=\operatorname{Spec}E\). Restriction of global functions defines the factorization \(C\to T\to V\). AF.3 proves that its first map is proper, smooth of relative dimension one, and has geometrically connected nonempty fibres. It computes the global functions after each algebraically closed field extension by AF.2, and identifies each geometric fibre with the factor selected by one primitive idempotent of \(E\). Its global-function algebra is that field, so the fibre is nonempty and connected. Properness follows from the closed graph and base change; smoothness follows from the open graph and the smooth product projection. The diagonal calculation underlying the open-graph assertion is as follows.

Here the diagonal's openness has a direct algebraic check. For a finite étale \(B\)-algebra \(E\), the kernel \(I\) of \(E\otimes_BE\to E\) is finitely generated and satisfies \(I/I^2=\Omega_{E/B}=0\). Write generators as \(x_i=\sum_j a_{ij}x_j\), with \(a_{ij}\in I\). The adjugate identity makes \(d=\det(\delta_{ij}-a_{ij})\) annihilate \(I\), and \(d\equiv1\pmod I\). Put \(e=1-d\). Then \(e\in I\), every \(x\in I\) satisfies \(x=ex\), and \(e^2=e\). Thus \(I=(e)\), so its closed subscheme is also the open \(D(1-e)\). This proves the diagonal assertion used above.

The product idempotents in the descended algebra split \(T=\coprod_iT_i\) and \(C=\coprod_iC_i\). Put \(Y_i=X_V\cap C_i\). The boundary is finite over \(V\), so also has zero-dimensional fibres over \(T_i\). Every geometric fibre of \(C_i\to T_i\) is a connected smooth curve. Its irreducible components cannot meet, since its regular local rings are domains; with finitely many components they would therefore form a disjoint open-and-closed decomposition. Connectedness forces one component. Deleting the finite boundary leaves a dense nonempty open in each such fibre. Density and nonemptiness descend from a faithfully flat algebraic closure of the residue field. This proves every assertion of (O.1) and the stronger version of (3.3).

For a smooth zero-dimensional generic component, its affine algebra over the generic field is finite étale. Finite presentation descends finiteness and its finite standard étale charts over a smaller open, so that component is finite étale there. It can be separated by descending the generic open-and-closed decomposition. This accounts for the zero-dimensional case without adding a curve calculation.

AG.1–AG.4 prove the normalization and perfect-field smoothness steps in O.1. AF.1–AF.4 prove the finite-map cohomology, arbitrary base change on the chosen neighbourhood, and connected-fibre factorization in O.4. AH.1–AH.3 prove the purely inseparable density, finite-boundary and generic-closure assertions. AH.4 verifies the hypotheses and proof chain for the finite diagrams in O.1–O.3, using the exact proved assertions of the limits lesson linked above.

Appendix P. The curve finiteness input

Let \(k\) be algebraically closed. We prove the curve assertion used in Proposition 3.2, keeping its geometric and cohomological inputs separate. Appendix Q supplies the Picard construction needed here, using the represented divisor charts of The Picard functor and the Picard scheme of a curve, Sections 2 and 4–6, with explicit coherent-cohomology inputs and a degree-\(g\) properness argument. It proves the following precise input for a smooth connected projective curve \(C\):

where \(J\) is a smooth proper connected commutative group scheme. The construction includes nonreduced parameter schemes. Q.1 proves the Euler-degree formula and the vanishing needed for the chart translations; Q.2 proves properness of the divisor parameter scheme; Q.3 supplies the finite cohomology complex for arbitrary parameters; and Q.4–Q.5 assemble the Picard scheme and prove properness. The high-degree Abel projective-bundle theorem and coherent Serre duality are not needed for this particular input. Their stronger statements elsewhere retain their own scope. The free comparison statements are [Stacks, Tags 0B9Z and 0BA0].

P.1. Division by an invertible integer

For any smooth proper connected commutative group scheme \(G/k\) of finite type and any positive integer \(n\) invertible in \(k\), multiplication \([n]:G\to G\) is surjective and étale. Its kernel has finitely many \(k\)-points. We prove precisely these assertions.

First consider a square-zero thickening \(T_0\subset T\) with ideal \(I\). The kernel of \(G(T)\to G(T_0)\), interpreted locally on \(T\), is

Here \(e\) is the identity. Indeed a map reducing to the identity has the same underlying topological image and hence lies in an affine neighbourhood of it. Subtract the identity's map on functions from the given map. The difference takes values in \(I\), and the product identity says exactly that it is a derivation, since \(I^2=0\). Conversely adding such a derivation gives a map on functions. This is Formally smooth, unramified and étale ring maps, Proposition 2.1, applied at the identity. The group operation on (P.2) is addition: its differential is linear on the two tangent factors, and its restrictions to either factor are the identity because multiplying by the identity does nothing. Consequently \([n]\) acts on (P.2) by the scalar \(n\).

Suppose \(x_0:T_0\to G\) and \(y:T\to G\) satisfy \([n]x_0=y|_{T_0}\). Smoothness of \(G/k\) supplies, locally on \(T\), a lift \(x\) of \(x_0\). The error \(\delta=[n]x-y\) lies in (P.2). Set

Then \([n]x'=y\). If two such lifts exist, their difference belongs to (P.2) and is killed by \(n\), so is zero. This uniqueness glues the local lifts. We have proved the infinitesimal lifting test for formal étaleness. The morphism is of finite presentation, since it is a morphism between finite-type schemes over the Noetherian field \(k\); hence it is étale by the finite-presentation criterion proved in Section 4 of the same algebra lesson. Successive square-zero thickenings also give the test for every nilpotent ideal.

AE.1 proves flatness for these étale charts over the Noetherian target, from the split presentation and the fibre-regular quotient argument. AD.2 then proves their openness. The map \([n]\) is also proper: its graph is closed in \(G\times_kG\) by separatedness, and the projection to the target is a base change of the proper structure map of \(G\). The base-change and composition assertions are proved in AE.2. Its image is therefore both open and closed, and contains the identity. Connectedness of \(G\) makes the image all of \(G\).

Each geometric fibre is a finite-type étale scheme over an algebraically closed field. AB.1 proves that it is a finite disjoint union of points, using the finite-presentation lifting criterion and the field-algebra calculation. In particular the fibre over the identity has finitely many points. Surjectivity implies surjectivity on \(k\)-points: the fibre over each \(k\)-point is nonempty, and its points have residue field \(k\) by the same classification. Thus

For \(n=1\) the kernel is the trivial group, as required. This proof uses neither a polarization nor an intersection-number calculation. Applied to the constructed \(J\), it gives all the division and torsion finiteness needed below.

P.2. Kummer on a proper smooth curve

AD.3 proves the degree-one identification by unit cocycles and line-bundle descent. Appendix R proves the higher vanishing by its function-field and valuation-sheaf calculation, as in The multiplicative group on a curve, Theorem 4.2. Together these give

Appendix R gives the function-field and valuation-sheaf deduction used here. Its calculation on finite generic fibres proves the needed higher-direct-image vanishing without the strict-local cohomology continuity theorem. It uses no Picard input. The divisor argument of the linked lesson also gives

X.2 constructs divisor representatives for every line bundle. X.3 proves finiteness of the morphism defined by a nonconstant rational function, and X.4 computes its fibre lengths, proving that principal divisors have degree zero. A point of \(C(k)\) has degree one. The kernel identification is (P.1).

X.1 proves that the global regular functions on \(C\) are \(k\), using finite-dimensional coherent sections and integrality. Thus \(\Gamma(C,\mathcal O_C)^\times=k^\times\), and the \(n\)-th-power map on this group is surjective. AD.4 proves the exact Kummer sequence by its explicit finite étale root cover. That sequence and (P.5) now give

and zero in degrees above two. The second equality in degree one follows because \(\mathbb Z\) has no torsion. The degree map identifies \(\operatorname{Pic}(C)/n\) with \(\mathbb Z/n\): if \(\deg L=na\), choose \(M\) of degree \(a\). Then \(L\otimes M^{-n}\) lies in \(J(k)\), and (P.4) supplies an \(n\)-th root of it. This proves the kernel assertion; a degree-one bundle proves surjectivity. Every group in (P.7) is therefore finite. Choosing a primitive \(n\)-th root identifies \(\mu_n\) with the constant sheaf \(\mathbb Z/n\). The choice is unnecessary for finiteness; the canonical degree map itself has coefficients \(\mu_n\). For the Kummer calculation, see [Stacks, Tag 03RN]. Only finiteness of torsion is used here.

P.3. The finite boundary and compact supports

Let \(U\) be a smooth connected affine curve. Over our algebraically closed field the normalization construction in O.1 gives an open immersion \(U\subset C\), with finite boundary \(D\), and a finite map from \(C\) to a projective curve. The completion \(C\) is projective as well as proper. To check this additional property, pull back the hyperplane bundle from that projective curve. Its coordinate-section opens pull back to affine opens because the map is finite; they cover \(C\). For a principal open \(D(b)\) inside one of these affine opens \(C_s\), the denominator-extension lemma gives \(a\geq0\) and a section \(u\) of the \(a\)-th tensor power with \(u/s^a=b\). The nonvanishing open of \(us\) is exactly \(D(b)\): its factor \(s\) excludes the complement of \(C_s\). Such principal opens form a basis, so the pulled-back bundle is ample by the affine-section-open definition. The denominator-clearing construction and affine-base very-ample-power theorem in Ample invertible sheaves, Lemma 2.1 and Theorem 4.2 embed a power into finite projective space. Properness makes this immersion closed. Since normalization preserves the connected smooth open and its closure is integral, \(C\) is a connected smooth projective curve.

The same completion construction applies to every smooth connected separated finite-type curve \(X\). Choose a nonempty affine open \(U\subset X\) and its smooth projective completion \(C\) just constructed. Y.6 and L.1 make the local ring at each closed point of \(X\) a DVR. The rational map \(X\dashrightarrow C\) extends over each such local spectrum by properness of \(C\). Finite presentation extends each local map to a neighbourhood, and separatedness of \(C\) makes these extensions agree on overlaps, since their maps on the common function field agree. They glue to a birational morphism \(X\to C\). Its closed fibres are proper closed subsets of the integral curve \(X\), hence finite; its generic fibre is one point. It is therefore quasi-finite and is separated because \(X\) is separated over the field. AW.4 identifies it with an open immersion into the integral closure of \(C\) in its unchanged function field. That integral closure is \(C\), since \(C\) is normal. Thus \(X\) is an open subscheme of \(C\) with finite boundary. If \(X\) is proper, the graph argument of AU.1 makes this open immersion proper, so its dense image is also closed and \(X=C\). Otherwise the boundary is nonempty, and X.4 makes its complement \(X\) affine. This proves both the projective and affine alternatives for a connected smooth curve.

Put \(\Lambda=\mathbb Z/n\), let \(j:U\hookrightarrow C\), and give \(D\) its reduced structure with inclusion \(i\). On geometric stalks the following sequence is either the identity sequence on \(U\) or the identity sequence on \(D\), so it is exact:

The definition of compact support and properness of \(C\) identify \(R\Gamma(C,j_!\Lambda)\) with \(R\Gamma_c(U,\Lambda)\). If \(r=|D|>0\), the cohomology sequence of (P.8) is

Also \(H_c^0(U,\Lambda)=0\) and \(H_c^q(U,\Lambda)=0\) for \(q>2\). Here a finite union of algebraically closed points has no positive étale cohomology, and finite direct image is exact, as proved in the valuation-sheaf calculation cited in P.2. If the boundary is empty, compact-support cohomology is simply the proper-curve cohomology from P.2. In either case all groups are finite. The same conclusions hold for finitely many connected components by taking their finite direct sum.

P.4. Constructible coefficients on any separated curve

Let \(X/k\) be separated of finite type and of dimension at most one, and let \(F\) be a constructible \(\mathbf F_\ell\)-sheaf, with \(\ell\) invertible in \(k\). Passing to the reduction does not change its étale topos or its compact-support cohomology, by finite topological invariance and proper-support composition.

There is an open \(U\subset X\) which is a finite disjoint union of smooth affine curves, on which \(F\) is finite locally constant, and whose complement \(Z\) is finite. Indeed, at the generic point of each one-dimensional reduced component its function field is separably generated over the perfect field \(k\), so the smoothness criterion supplies a dense smooth open. Intersect it with the constructible sheaf's open lisse stratum, delete the intersections with other components, and choose a nonempty affine open inside it. Its complement in that component is zero-dimensional and closed, hence finite. There are finitely many components; any zero-dimensional components can be included in \(Z\). These choices give the stated disjoint open union. If there are no curve components, take \(U\) empty.

On a connected component of \(U\), the finite-monodromy construction N.2–N.3 and the elementary subgroup argument in Proposition 3.2 give a finite étale cover \(g:U'\to U\) of degree \(e\) prime to \(\ell\) for which \(g^*F\) has a finite filtration with constant \(\mathbf F_\ell\) quotients. This subgroup and invariant-vector argument uses only finite group theory, so its occurrence later in the proof of Proposition 3.2 creates no dependence on that proposition's finiteness conclusion. The cover is affine, smooth and of dimension one, since it is finite étale over \(U\). P.3 and successive cohomology sequences prove finiteness and vanishing above two for \(H_c^q(U',g^*F)\).

The unit and the sum over the finite étale fibres define

This identity is checked after splitting the cover étale locally and hence on all geometric stalks. Multiplying the trace by \(e^{-1}\) gives a retraction. Finite direct image is exact and agrees with proper-support image; proper-support composition therefore makes \(H_c^q(U,F)\) a direct summand of \(H_c^q(U',g^*F)\). It too is finite and zero above two.

Finally apply compact-support excision to \(U\subset X\). The term on the finite complement \(Z\) is a finite group in degree zero and zero in higher degrees. The long exact sequence proves the claimed finiteness and vanishing for \(X\). Filtering a finite \(\ell\)-primary coefficient sheaf by powers of \(\ell\) proves the same assertion for it. Finite truncation triangles then give finite compact-support cohomology for every bounded complex with such constructible cohomology; a sheaf range \([a,b]\) gives the range \([a,b+2]\) on a curve. For a proper curve these are ordinary cohomology groups. Every reduction uses only the independently constructed compact-support operations and the explicit curve and group arguments above, before general direct-image constructibility is proved. At the smooth-curve scope, AK.4–AK.5 give a direct proof using finite normalization and a projective completion, without a general proper-support composition input.

Appendix Q. Divisor charts and the Picard input

Fix an algebraically closed field \(k\), a smooth connected projective curve \(C/k\), and a point \(p\in C(k)\). Put \(g=\dim_k H^1(C,\mathcal O_C)\). We construct the smooth proper connected group required in (P.1), retaining arbitrary parameter schemes. The argument uses the finite-algebra construction of the curve's divisor functor, not projectivity of a general Quot scheme.

Q.1. Euler degree and vanishing at one point

Choose two hyperplanes in a projective embedding of \(C\) whose intersections with \(C\) are disjoint. To do this, choose the first not containing \(C\); its intersection is finite. Since \(k\) is infinite, choose the second avoiding those finitely many points and not containing \(C\). Their complements give a cover by two affine opens with affine intersection. W.1–W.2 prove that the ordered Čech complex computes quasi-coherent cohomology in degrees zero and one only. This remains true after any field extension \(K/k\), and after any change of parameter ring.

W.3–W.4 prove finite coherent cohomology by the projective-space monomial calculation, finite quotients by twists and descending induction, then a common affine-cover complex for a closed projective subscheme. They give finite-dimensional cohomology here and finite modules over any Noetherian parameter ring. No duality theorem enters it. Flat field extension tensors the affine-cover complex term by term, so preserves cohomology dimensions. In particular \(H^0(C_K,\mathcal O)=K\) and \(h^1(C_K,\mathcal O)=g\).

For a closed point \(x\) of \(C_K\) and a line bundle \(L\), the discrete valuation local ring gives

The last sheaf has only degree-zero cohomology, of dimension \([K(x):K]\). Additivity of the finite Euler characteristic therefore gives \(\chi(L)-\chi(L(-x))=[K(x):K]\). X.2 proves that every line bundle has a rational section and is \(\mathcal O(D)\) for a finite divisor \(D\), including over the extension field \(K\). Iterating (Q.1) forward and backward over the positive and negative coefficients proves

Here degree is the divisor degree, with residue-field multiplicities. In particular this formula proves that the divisor and Euler-characteristic definitions agree. Since cohomology commutes with field extension, the degree does too. It also gives the inequality \(h^0(L)\geq\deg L+1-g\). We have proved the portion of curve Riemann–Roch needed in this appendix directly from point exact sequences; no identification of \(H^1\) with a dual space is required.

For \(m\geq g+1\), (Q.2) gives \(h^0(C,\mathcal O(mp))\geq2\). Choose a nonconstant element of that space. It is a rational function with its only pole at \(p\): having no pole would make it a constant regular function on the proper curve. It defines a morphism \(C\to\mathbf P^1\), since at a discrete valuation ring either the function or its inverse is regular. X.3 proves this map is finite by its projective graph and W.4 coherent finiteness; X.4 proves surjectivity and the exact fibre degrees. The inverse image of \(\mathbf A^1\) is exactly \(C\setminus\{p\}\), which is therefore affine. This argument works with any point of \(C(k)\). The same nonconstant finite map is surjective onto \(\mathbf P^1\); its fibres over the infinitely many \(k\)-points show that \(C(k)\) is infinite.

Let \(L\) now be any line bundle on \(C_K\). With \(U_K=C_K\setminus\{p_K\}\) and inclusion \(j\), pole orders give an equality of quasi-coherent sheaves

Locally this is ordinary localization at an equation of \(p_K\): each fraction has a finite pole order. Compute both sides on the two-affine cover above. Filtered colimits commute with its finite products, kernels and cokernels; the intersections of these affine opens with \(U_K\) are affine as well. Thus the colimit of \(H^1(C_K,L(mp_K))\) is \(H^1(U_K,L|_{U_K})=0\). The transition maps on \(H^1\) are surjective by (Q.1). Their finite dimensions decrease and eventually stabilize; in the stable range the surjections are isomorphisms. Their zero colimit forces that stable dimension to be zero. Consequently

This proves the twisting vanishing used below over every extension field, including imperfect extensions of \(k\).

Q.2. The finite-divisor parameter scheme

Write \(H_d\) for the functor of closed subschemes of \(C_T\) finite locally free of degree \(d\) over \(T\). U.3 proves that these families are relative effective Cartier divisors over arbitrary parameter schemes, by lifting fibre generators and descending the finite algebra matrices. The finite-divisor charts are constructed in Hilbert and Quot schemes, Sections 5 and 7.

For clarity, the representability used here is the elementary part of that lesson. Choose \(d+1\) distinct points of \(C(k)\). Any degree-\(d\) support misses one; Q.1 makes its complement affine. These finitely many affine opens suffice to cover the divisor functor, because the locus where a finite family lies inside a specified open is open: its complement is the finite image of the part of the family outside that open. On an affine curve with generators \(z_1,\ldots,z_s\), a rank-\(d\) quotient algebra is covered by the choices of a basis of monomials of degree at most \(d-1\), containing \(1\). These monomials span over every field: until their span is the whole algebra, multiplication by the generators increases its dimension. There can be at most \(d-1\) strict increases.

For a fixed monomial basis, multiplication by each \(z_i\) is a matrix. The equations that the matrices commute, satisfy the curve's relations, and send the vector of \(1\) to the chosen monomial basis vectors are finite polynomial equations. Conversely those equations make the resulting free module a cyclic quotient algebra: the kernel of evaluation at the vector of \(1\) is an ideal. Overlaps invert a change-of-basis determinant and conjugate the matrices. This is the full construction in Proposition 5.1 of the linked lesson, including finite presentation and the gluing maps. It gives a scheme \(H_d\) of finite type. For \(d=0\) it is \(\operatorname{Spec}k\).

We supply the separation and properness needed here. For two families \(D_1,D_2\) over \(T\), containment \(D_1\subset D_2\) means that the map

is zero. Both sheaves are finite locally free of rank \(d\): \(D_1/T\) is finite locally free, and the restricted ideal is invertible. They and their map commute with arbitrary base change. Locally the zero condition is exactly the vanishing of all entries of a matrix, so is represented by a closed subscheme of \(T\). On that subscheme the containment is equality: it gives a surjection between the two finite locally free rank-\(d\) algebras, whose determinant is invertible at every point. Thus the diagonal of \(H_d\) is closed, proving separation.

Adding the \(d\) graph divisors gives a morphism \(C^d\to H_d\); U.3–U.4 prove that their product ideal defines a finite locally free family of degree \(d\). On a geometric fibre every finite divisor is a sum of points with multiplicities, by the discrete valuation description, so the map is surjective after every base change. AE.2 proves properness of \(C^d\) from the projective embedding and the graded closed-image calculation. For any \(T\) and closed subset \(B\subset H_d\times T\), its inverse image in \(C^d\times T\) is closed. Its image in \(T\) is closed by that properness, and equals the image of \(B\) by surjectivity. Hence \(H_d\) is universally closed over \(k\). Finite type and separation make it proper. AE.3 proves that \(C^d\) remains integral after every field extension; the same surjection therefore proves geometric irreducibility of \(H_d\).

The explicit etale charts in V.1–V.3 prove smoothness without a characteristic restriction. V.4 also proves the following local coefficient description. At \(D=\sum_i d_i x_i\), choose formal parameters \(u_i\). Over a local Artinian test algebra \(R\) with residue field \(k\), idempotents split a finite flat deformation into its distinct support clusters. In the cluster of length \(d_i\), Nakayama makes \(1,u_i,\ldots,u_i^{d_i-1}\) a basis. It has one unique relation

Conversely every such relation defines a free quotient with that basis. If \(\mathfrak m_R^N=0\), it implies \(u_i^{d_iN}=0\); hence formal series evaluate by finite sums, so the polynomial and formal descriptions agree. Therefore

with \(d\) variables. V.4 constructs the inverse maps giving this power-series description, including its separate support clusters. Smoothness and dimension \(d\) follow independently from V.3: a neighbourhood of every closed point has an etale coefficient map to \(\mathbf A^d_k\), and those neighbourhoods cover \(H_d\). Thus \(H_d\) is smooth, proper and geometrically integral. Only these properties of divisors are used below; no division by \(d!\) has occurred.

Q.3. Cohomology over arbitrary parameter rings

For a line bundle \(L\) on \(C_A\), with \(A\) any \(k\)-algebra, there is locally on \(\operatorname{Spec}A\) a complex of finite free modules

which computes the cohomology of \(L_B\) after every \(A\)-algebra change \(B\).

Here is a proof specialized to this fixed curve. Write \(A\) as the filtered union of its finitely generated \(k\)-subalgebras \(A_i\). The invertible sheaf descends to an invertible sheaf \(L_i\) on \(C_{A_i}\). On the fixed finite affine cover, descend its finite module presentations, its dual, the evaluation isomorphisms, the gluing maps and their inverses. Each involves finitely many coefficients. The equations for the inverse maps and triple-overlap identities hold at a sufficiently late stage. These explicit finite data give the descent; the finite-diagram arguments are also proved in O.2–O.3's limit provider. Because the curve is fixed over a field, \(C_{A_i}\) is \(A_i\)-flat, and the descended sheaf is invertible, so it is automatically \(A_i\)-flat. No descent of flatness for a general coherent sheaf is needed.

Over Noetherian \(A_i\), the two-affine Čech complex has flat \(A_i\)-modules in degrees zero and one. Flatness follows by tensoring an injection and using exact affine sections; it does not require those section modules to be finite over \(A_i\). W.4 proves both the stated flatness and finite cohomology over \(A_i\), using the affine comparison in W.2. U.1 constructs a finite replacement for precisely this two-term flat complex: adjoin finitely many representatives of its degree-one cohomology and take the kernel of the resulting surjection. That kernel is finite and flat, hence locally finite free, and the comparison remains a quasi-isomorphism after every tensor product. Tensor that comparison with \(A\), and then with any \(B\). The affine cover and all its section modules base change term by term, proving (Q.8). The finite model is taken on the free neighbourhoods supplied by U.1.

Two consequences now follow by finite matrices. First, \(\chi(L_t)=\operatorname{rank}K^0-\operatorname{rank}K^1\) is locally constant. Second, if \(h^0(L_t)=1\) and \(h^1(L_t)=0\) at a point, a full-size minor of \(d\) is invertible nearby. The map is surjective there and splits; its kernel is a line bundle \(N\). The complex is the direct sum of \(N[0]\) and a two-term identity complex. This decomposition survives every base change. Thus the condition

defines an open subfunctor, and on it \(f_*L=N\) with arbitrary-base-change compatibility. These are statements about actual parameter schemes, including nilpotents, not just their field-valued points.

Q.4. A represented chart and its translates

For every \(T/k\), the affine-cover calculation and exact tensoring over \(k\) give \((C_T\to T)_*\mathcal O_{C_T}=\mathcal O_T\). Hence every automorphism of a line bundle comes from a unique base unit. Normalize a line bundle by tensoring with the inverse pullback of its restriction to \(p_T\), and retain its canonical trivialization there. An automorphism respecting that trivialization is the identity.

Rigidified bundles form an fppf sheaf by AD.1–AD.2 and AD.4. The argument descends finite presentation and rank one over arbitrary rings, constructs finite affine refinements of fppf covers over every parameter scheme, and descends both the bundle and its trivialization. Uniqueness of rigidified overlap isomorphisms gives the cocycle. Thus the relative Picard sheaf is the functor of rigidified bundles on every \(T\), with the normalization and quotient identification proved in AD.4.

Let \(F\) be the open subfunctor (Q.9). Its elements have degree \(g\) by (Q.2). Q.3 makes its inclusion representable by open immersions. It is represented by

The functorial inverse is worth specifying. For \(L\in F(T)\), evaluate the line bundle \(N=f_*L\) to obtain \(f^*N\to L\). It is nonzero on every curve fibre. U.4 proves that its zero scheme is a relative effective Cartier divisor, finite locally free of degree \(g\) over \(T\), including non-Noetherian parameter schemes. Its associated bundle is \(L\otimes f^*N^{-1}\). Normalizing at \(p_T\) cancels this base factor and recovers the original rigidified bundle. In the reverse direction, the canonical section of \(\mathcal O(D)\) spans its one-dimensional space of sections on every fibre. Q.3 makes \(\mathcal O_T\to f_*\mathcal O(D)\) an isomorphism, and evaluation recovers exactly \(D\). These constructions commute with all base changes, proving (Q.10) as an isomorphism of functors.

Every field-valued Picard point can be translated into this chart by a bundle defined over \(k\). For \(L\) on \(C_K\), choose \(m\) so large that (Q.4) kills its first cohomology and (Q.2) makes \(h^0\geq1\). If \(h^0>1\), take a nonzero section and choose \(x\in C(k)\) outside its finite zero divisor on \(C_K\); infinitely many such \(k\)-points are available. The evaluation in (Q.1) is surjective, so twisting by \(-x_K\) reduces \(h^0\) by one and leaves \(h^1=0\). Repeat until \(h^0=1\). All twists are the pullback of

a bundle defined on \(C\).

Translate the represented open \(W\) by all elements of \(\operatorname{Pic}(C)\). Pairwise intersections are represented opens in both charts and identify the same subfunctor, so their identifications satisfy the cocycle identity. Glue them. They cover the Picard functor on every parameter scheme: at each parameter point apply (Q.11) to its residue field, and then take the inverse image of the resulting open chart. This is the proof of the represented-open gluing criterion in Section 4 of the Picard provider. We obtain a smooth group scheme \(P\), locally of finite type and of dimension \(g\), representing all rigidified bundles. The universal rigidified line bundle is the one corresponding to the identity map of \(P\).

Q.5. Proper degree pieces from degree \(g\) divisors

The group scheme \(P\) is separated. Its rational identity has no proper specialization: in an affine neighbourhood of a proposed specialization, the identity point corresponds to a maximal ideal, since evaluation is a surjective \(k\)-algebra map to \(k\). Thus the point is closed. Evaluation also identifies its inclusion with a closed immersion on that affine neighbourhood; the complement of the point supplies the empty closed immersion on the remaining target open. Its pullback under \((a,b)\mapsto a-b\) is exactly the diagonal as a functor on schemes, so the diagonal is a closed immersion.

By Q.3 and (Q.2), degree is locally constant in a family. Apply this to the universal bundle to decompose

Each piece is open and closed. Tensoring by \(\mathcal O_C(rp)\) gives an isomorphism \(P^d\to P^{d+r}\), and tensor product adds degrees. The universal divisor gives the Abel map \(H_g\to P^g\). It is universally surjective: over any extension field, (Q.2) gives at least one section of every degree-\(g\) bundle. Its zero divisor has degree \(g\) and maps to that bundle. Checking geometric points proves surjectivity after every base change.

The image of the quasi-compact scheme \(H_g\) is all of \(P^g\), so \(P^g\) is quasi-compact. It is locally of finite type, hence of finite type. For any base \(T\) and closed subset of \(P^g\times T\), its inverse image in \(H_g\times T\) is closed and has closed image in \(T\), because \(H_g\) is proper. Universal surjectivity identifies that image with the original closed subset's image. Thus \(P^g\) is universally closed, and its separation and finite type make it proper. The same surjection makes it geometrically irreducible; its smoothness makes it geometrically reduced. Translations give these properties for every \(P^d\).

In particular \(J=P^0\) is smooth, proper and geometrically connected. It is the identity component, because it is connected, open and closed, and all degree pieces are disjoint opens. Its group of \(k\)-points consists exactly of degree-zero line bundles, proving (P.1). The argument includes \(g=0\): then \(H_g\) is a point and the degree-zero piece is a point. No high-degree Abel-fibre formula, coherent Serre duality, or projectivity theorem for a general Quot scheme is needed for this construction. The stronger versions of those results are unaffected; only the precise Picard input for curve finiteness has been proved here.

Appendix R. Function fields and rational units

For Appendix P we need the vanishing of positive-degree unit cohomology over function fields, followed by a calculation on a smooth curve. The finite stages themselves suffice for the second step. Throughout this appendix, \(k\) is algebraically closed and \(F/k\) has transcendence degree one; finite generation of \(F\) is not assumed.

R.1. A polynomial count over the constant field

We first recall precisely the elementary dimension input. Fewer homogeneous equations of positive degree than variables have a common nonzero zero over \(k\). Indeed, if \(r<N\) equations had only the origin as zero, the strong Nullstellensatz proved in Y.5 would make the radical of their ideal equal to \((Z_1,\ldots,Z_N)\). The height bound (Y.7) makes this prime have height at most \(r\), whereas the chain of coordinate primes has length \(N\). Zero equations can be discarded.

Let \(f\) be a homogeneous form of degree \(d>0\) in \(n>d\) variables over \(F\). Choose a transcendental \(t\in F\). The field \(F_0\) generated over \(k\) by \(t\) and the finitely many coefficients of \(f\) is finite over \(k(t)\), say of degree \(e\). Choose a \(k(t)\)-basis \(w_1,\ldots,w_e\) of \(F_0\). This argument allows inseparability. For an integer \(m\geq0\), substitute

Expand the finitely many products consisting of a coefficient of \(f\) and \(d\) basis elements in that basis. Their rational coordinates admit a common nonzero polynomial denominator \(D(t)\). Let \(B\) bound the degrees of the resulting polynomial numerators. Both \(D\) and \(B\) are fixed before choosing \(m\). Consequently

where the \(h_{ib}\) are homogeneous of degree \(d\), or zero. Thus there are at most \(e(dm+B+1)\) equations in \(ne(m+1)\) scalar variables. Choose \(m\) so that

The preceding dimension argument gives a nonzero solution \(z\). Transcendence of \(t\) and linear independence of the \(w_i\) imply that the resulting tuple \(x\) is nonzero. Equation (R.2) gives \(f(x)=0\). This proves that \(F\) is \(C_1\). It also proves the same assertion for every algebraic extension of \(F\), since that extension still has transcendence degree one over \(k\). No passage of Brauer groups or cohomology through a field limit is used. This is the finite-coefficient form of the proof in Brauer groups and Tsen's theorem, Section 5; see also [Stacks, function fields].

R.2. From forms to degree-two cohomology

Z.1 proves the finite-dimensional simple-algebra classification and scalar extension used here. Z.2 constructs a finite separable splitting field over the infinite fields in this appendix, using a trace Gram determinant to choose an element with distinct eigenvalues. Passing to its finite normal closure gives a Galois splitting field, including in positive characteristic over imperfect fields.

For a central division algebra \(A/F\) of degree \(a\), Z.2 constructs its reduced norm as a degree-\(a\) homogeneous polynomial over \(F\) in \(a^2\) coordinates, and proves that it is nonzero exactly on invertible elements. If \(a>1\), R.1 produces a nonzero element of norm zero, contradicting division. Thus \(A=F\), and Z.1 makes every central simple algebra a matrix algebra. In particular \(\operatorname{Br}(F)=0\).

Z.3 proves the finite-dimensional semilinear descent needed next. Z.4 associates a descended matrix algebra to each finite scalar two-cocycle. Once that algebra is split, its invariant matrix units make the semilinear operators scalar, and their multiplication identity explicitly expresses the cocycle as a coboundary. The full identification with the Brauer group is Brauer groups and Tsen's theorem, Theorem 3.1. The local scalar calculation and the division-algebra argument already prove, for every field \(L\) algebraic over \(F\),

Here \(L^s\) denotes a separable closure. The sign convention for the scalar boundary has no effect on this vanishing statement.

R.3. Additive contraction and the Sylow test

We use continuous group cohomology. AA.2 constructs enough injective discrete modules and proves that continuous homogeneous cochains compute the derived functors of invariants. It also proves the finite-quotient comparison (AA.4) and compatibility with filtered colimits of coefficients. AB.1–AB.2 supply the field-site classification and the exact equivalence identifying these groups with étale cohomology, including the additive and multiplicative coefficient modules.

Here is a direct additive calculation valid for every field \(L\). If \(E/L\) is finite Galois with group \(Q\), the map \(\operatorname{Tr}(b)=\sum_{g\in Q}g(b)\) is nonzero: otherwise it would be a nontrivial linear relation between distinct field automorphisms. The minimal-relation proof of their independence is given in Z.3 and works over every field. Its values lie in \(L\), so scaling a value which is nonzero supplies \(a\in E\) with \(\operatorname{Tr}(a)=1\).

For an equivariant homogeneous cochain \(c:Q^{r+1}\to E\), with omission differential \(d\), define, for \(r\geq1\),

This is equivariant: substitute \(u=hv\) when translating every argument by \(h\), and use both \(u(a)=h(v(a))\) and equivariance of \(c\). In \(ds+sd\), all terms omitting one of the original arguments cancel in opposite pairs. The terms omitting the newly inserted argument sum to \(\sum_u u(a)c=c\). Therefore \(ds+sd=1\) in positive degrees, and \(H^r(Q,E)=0\) there. Finite-quotient comparison, with \((L^s)^U=E\), now gives

This supplies additive acyclicity directly, without a normal-basis existence theorem. In characteristic \(p\), the polynomial \(T^p-T-b\) has derivative \(-1\) and splits over a separably closed field. Thus

is exact. Equation (R.6) and its long exact sequence give \(H^r(G_L,\mathbf F_p)=0\) for \(r\geq2\).

Now put \(G=G_F\), take the Sylow pro-\(\ell\) subgroup \(P\) constructed in AA.3, and let \(L=(F^s)^P\). AA.1 proves that \(F^s\) is a separable closure of \(L\) and \(G_L=P\) with its induced topology. The field \(L/k\) still has transcendence degree one. If \(\ell\ne\operatorname{char}k\), the action of \(P\) on \(\mu_\ell\) is trivial: its finite image has \(\ell\)-power order dividing \(\ell-1\). Scalar Hilbert 90, proved in Z.3, makes the Kummer long exact sequence inject \(H^2(L,\mu_\ell)\) into \(H^2(G_L,(L^s)^\times)\), which vanishes by Z.4. Hence

A choice of primitive root supplies the first isomorphism. For \(\ell=p=\operatorname{char}k\), (R.7) gives the same vanishing. In particular this reasoning never needs to identify \(H^2(L,\mu_\ell)\) with a limit of cohomology groups of smaller fields.

AA.5 proves the passage from this single coefficient to all torsion coefficients. A finite nonzero \(\ell\)-primary \(P\)-module has a filtration by trivial \(\mathbf F_\ell\)-lines, by orbit counting on its \(\ell\)-torsion and iteration on quotients. Long exact sequences and filtered unions give degree-two vanishing for every discrete \(\ell\)-primary module. The acyclic continuous-function module (AA.3) and its successive quotients give all higher degrees. Restriction from \(G\) to \(P\) is injective on \(\ell\)-primary cohomology by (AA.10): a cocycle and a primitive of its restriction descend to one finite quotient, where the explicit prism and coset-sum formulas (AA.7)–(AA.9) multiply the cocycle by the prime-to-\(\ell\) index. Consequently

For general torsion \(T\), take its primary parts and their finite direct sums; filtered-cochain comparison justifies the union.

R.4. The multiplicative module is not torsion

Write \(M=(F^s)^\times\) additively for this paragraph. Let \(T\subset M\) be its torsion subgroup, \(N=M/T\), and \(V=N\otimes_{\mathbb Z}\mathbb Q\). The group \(N\) is torsion free: if a multiple of a class is torsion in \(M\), another multiple annihilates its representative. Thus

are exact sequences of discrete continuous modules, with \(V/N\) torsion. Discreteness of the rationalization follows by intersecting the open stabilizers of the finitely many numerators in a representative. Positive-degree cohomology of \(V\) vanishes by AA.5: the finite homogeneous insertion sum annihilates finite-group cohomology by the group order, and (AA.4) therefore makes each positive profinite class torsion. With rational coefficients the same cohomology is a rational vector space, so it is zero.

The second sequence of (R.10), followed by (R.9), gives \(H^r(G_F,N)=H^{r-1}(G_F,V/N)=0\) for \(r\geq3\). The first then gives \(H^r(G_F,M)=0\) for \(r\geq3\). Degree two is (R.4), and degree one is scalar Hilbert 90, whose finite-value proof works for arbitrary fields. The field-topos comparison (AB.5) therefore proves

The separate degree-two argument is necessary: torsion cohomological dimension alone would not give that degree for \(M\).

R.5. Calculate before taking a strict-local limit

Let \(X/k\) be a smooth connected separated curve, \(K=k(X)\), and \(j:\operatorname{Spec}K\to X\). A quasi-compact étale scheme \(U\to X\) of finite presentation has generic fibre

where the \(E_i/K\) are finite separable extensions; the empty fibre is allowed. AB.1 proves this finite étale algebra description over a field. The restriction of \(\mathbf G_{m,K}\) to the étale slice is its units sheaf, since AB.3 identifies the slice with the small étale site of \(U_K\). Each \(E_i/k\) has transcendence degree one. Equation (R.11) and exactness of finite products therefore give \(H^r(U_K,\mathbf G_m)=0\) for \(r>0\).

The slice-resolution formula (AB.8) identifies \(R^rj_*\mathbf G_{m,K}\) with the sheafification of \(U\mapsto H^r(U_K,\mathbf G_m)\). The presheaf is zero on the quasi-compact finite-presentation basis, so its associated sheaf is zero. AB.3 also proves that inverse image is exact and direct image preserves injectives. An injective resolution on \(\operatorname{Spec}K\) therefore becomes an injective resolution of \(j_*\mathbf G_{m,K}\). Taking sections and using (R.11) yields

It remains to compare rational units with regular units. AB.4 proves preservation of a local parameter on étale curve neighbourhoods. AB.5 uses those parameters to lift every divisor section locally and proves the kernel calculation, giving the valuation sequence also proved in The multiplicative group on a curve, Sections 1–2:

AB.4 also proves the degree-zero strict-local description: the pointed local-ring colimit is a discrete valuation ring with the common uniformizer, and the quotient of its fraction-field units by its ring units is \(\mathbb Z\). The local-lift proof in AB.5 establishes (R.14) directly on neighbourhoods. Neither argument uses a higher-cohomology limit comparison.

For a single closed point \(x\), the stalk of \(i_{x*}A\) is \(A\) at \(x\) and zero elsewhere. A quasi-compact étale neighbourhood has finite fibre over \(x\); select the pointed component and remove the other fibre points. Away from \(x\) one can exclude \(x\). For exactness and cohomology, AB.5 gives the section formula \((i_{x*}A)(U)=A^{U_x}\), so direct image is exact. AB.3 makes it preserve injectives, and AB.2 identifies the étale topos of algebraically closed \(k\) with sets. Thus \(H^r(X,i_{x*}\mathbb Z)=H^r(k,\mathbb Z)=0\) for \(r>0\).

Appendix S proves directly that the infinite point sum in (R.14) has zero positive-degree cohomology. Its finite-cover contractions preserve support in the base points; the signed total-complex argument supplies the comparison with derived sections. In particular this deduction requires neither general hypercover comparison nor a theorem interchanging arbitrary filtered coefficient colimits with cohomology. The long exact sequence of (R.14), constructed in T.4, uses that vanishing and (R.13) to prove

which is the higher-degree assertion required in (P.5). Line-bundle descent supplies its degree-one Picard identification. The construction retains the algebra, continuous-cochain and sheaf foundations explicitly; neither a strict-local higher-continuity assertion nor a normal-basis theorem is an additional input to this deduction.

Appendix S. Finite covers and the infinite divisor sheaf

The sum of all closed-point valuation sheaves in (R.14) is infinite. Its acyclicity does not follow merely by adding the separate point calculations: derived sections need not commute with an arbitrary infinite sum. We prove the required assertion using finite families of étale covering maps, with explicit contractions and a finite elimination argument for the comparison with derived sections.

S.1. The finite-cover basis and the sections to be calculated

Let \(X\) be as in R.5. Use the basis \(\mathcal B\) of quasi-compact separated étale schemes of finite presentation over \(X\). AC.4 proves that it is a basis, that finite fibre products over its objects stay in it, and that every cover refines to a finite covering family in it. The proof uses the open-diagonal and cancellation calculation of AB.3, the local parameter calculation of AB.4, and the Noetherian image and flat-openness arguments of AC.1–AC.3. It also proves the same finite-cover assertion after base change to any basis object. Thus these finite covers generate the restricted site. T.1 identifies its sheaves and cohomology with those on the full small étale site.

For each closed point \(x\) of \(X\), let \(A_x\) be an abelian group, and put

Every closed point of a finite-type \(k\)-scheme has residue field \(k\), by the Nullstellensatz, since \(k\) is algebraically closed. For \(U\in\mathcal B\), its fibre \(U_x\) is a finite discrete set of such points. The direct image of \(A_x\) has sections \(A_x^{U_x}\). We claim that the sheaf sum in (S.1) has the exact section formula

where \(f:U\to X\). Closed points of \(U\) map to closed points of \(X\): their residue field is \(k\), and the image of a \(k\)-point in a finite-type \(k\)-scheme is closed. Thus the second indexing is precisely the disjoint union of the finite fibres in the first one.

To prove the first equality, take a finite cover of \(U\). A matching family in the presheaf sum uses finitely many \(x\)'s on each member, hence finitely many in total. It therefore belongs to a fixed finite sum of the sheaves \(i_{x*}A_x\), where the sheaf axiom gives a unique glue. Equality can be checked in the same finite sum. Hence the presheaf sum already satisfies the sheaf axiom for the finite-cover topology on \(\mathcal B\). That topology generates the basis site, so its sheafification has exactly these sections. This is only a degree-zero argument; no higher-cohomology colimit assertion has been used.

S.2. A signed comparison using finitely many diagonal terms

We record the sheaf-cohomology tools needed to pass from a finite-cover calculation to derived groups. T.1–T.4 supply their abelian-sheaf, injective-resolution and comparison foundations explicitly. Choose a resolution \(F\to I^\bullet\) in nonnegative degrees. The exact slice adjunction (AB.7) proves that restriction preserves injectives: its left adjoint sheafifies the direct sum over arrows into the slice, and finite sums of local lifts can be lifted on a common refinement. Thus \(H^n(U,F)\) is the cohomology of \(I^\bullet(U)\) and agrees with cohomology on the slice over \(U\).

For a finite cover \(\mathcal V=\{V_i\to U\}_{i=1}^m\), form the full Čech complex, retaining repeated indices:

The differential \(d_h\) is the alternating sum of restrictions obtained by omitting an index. Its square is zero: the two orders of omitting any pair give opposite signs.

The augmented Čech complex of an injective sheaf is exact. Here is the local chain argument behind that assertion. Write \(P_V\) for the free abelian sheaf on the representable \(V\), so \(\operatorname{Hom}(P_V,I)=I(V)\). Form the augmented chain complex of the \(P_{V_{i_0\ldots i_p}}\), ending in \(P_U\). A sheaf cycle is locally a finite sum of arrows into these products. After a common covering refinement, every arrow into \(U\) occurring in that sum has a chosen lift to some \(V_i\). Inserting this lift first in each tuple defines a chain contraction: the face deleting the inserted lift is the original tuple, and all other faces cancel against insertion after taking the boundary. It respects each arrow into \(U\), hence each of the finitely many supports. Equalities introduced by sheafification hold after a further cover. The chain complex is therefore exact as a sheaf complex. Applying \(\operatorname{Hom}(-,I)\) is exact for injective \(I\), proving the assertion, including the augmentation. This is the free-chain proof in Section 3, Lemma 3.1 of the cohomology lesson.

Now use the double complex

Here \(d_v\) is the resolution differential, which commutes with restrictions. The sign makes \(d_{\rm Tot}^2=0\). The total group in degree \(n\) has exactly the \(n+1\) diagonal summands \(D^{p,n-p}\). The restriction augmentation

is a quasi-isomorphism, as the following direct elimination proves.

For surjectivity on cohomology, let \(z\) be a total cocycle of degree \(n\). If its highest nonzero horizontal column is \(r>0\), with component \(z^{r,n-r}\), the cocycle equation in column \(r+1\) says \(d_hz^{r,n-r}=0\). The augmented row is exact, so write this component as \(d_hb^{r-1,n-r}\). Subtracting \(d_{\rm Tot}b\) removes column \(r\) and changes only column \(r-1\). Repeating leaves a representative in column zero. Its horizontal differential is zero, so row exactness makes it the restriction of a unique \(a\in I^n(U)\). Its vertical differential is zero, and injectivity of the row augmentation gives \(d_va=0\). This is a preimage under (S.5).

For injectivity, suppose a restricted global cocycle is \(d_{\rm Tot}b\). If \(b\) has highest column \(r>0\), its horizontal differential there is zero, since \(d_{\rm Tot}b\) has only column zero. Write that component as \(d_hc^{r-1,q}\) and replace \(b\) by \(b-d_{\rm Tot}c\); the total differential is unchanged and the highest column decreases. At the end, \(b\) lies in column zero and has zero horizontal differential. It is the restriction of some \(v\in I^{n-1}(U)\), and the original global cocycle equals \(d_vv\), by injectivity of restriction into the Čech row. In degree zero, there are no boundaries and the augmentation is already injective. These finite procedures prove (S.5) in every degree, without a spectral-sequence convergence or hypercover theorem.

S.3. When these Čech contractions imply derived vanishing

Suppose an abelian sheaf \(F\) has \(\check H^p(\mathcal V,F)=0\) for \(p>0\), for every finite basis cover of every \(U\in\mathcal B\). Then

We prove this simultaneously over all basis objects by induction on \(n\). Represent a class by a global resolution cycle \(a\in I^n(U)\). Exactness of the resolution as a sheaf complex means that it is locally a boundary. Choose a finite basis cover and sections \(b_i\in I^{n-1}(V_i)\) with \(d_vb_i=a|_{V_i}\). This is permitted by S.1's cofinality. Under (S.5), the class has initial representative in bidegree \((0,n)\). Subtracting the total differential of \((b_i)\in D^{0,n-1}\) removes that component.

Suppose the current lowest nonzero column is \(p\), with \(0<p<n\). Its component has vertical degree \(q=n-p\), strictly between zero and \(n\). The total cocycle identity makes it a vertical cycle. Every intersection in (S.3) is a basis object, and its degree-\(q\) cohomology is zero by induction. The product in (S.3) is finite, so choose primitives component by component. Multiplying by \((-1)^p\) gives \(b^{p,q-1}\) with \((-1)^p d_vb^{p,q-1}=z^{p,q}\). Subtract \(d_{\rm Tot}b\); this removes column \(p\) and changes only column \(p+1\). After finitely many steps the representative lies in \((n,0)\).

The vertical cycle condition now puts it in \(\check C^n(\mathcal V,F)\), because \(F\) is the kernel of \(I^0\to I^1\). Its horizontal differential is zero. The assumed Čech vanishing writes it as \(d_hc\), for \(c\in\check C^{n-1}(\mathcal V,F)\). Such a \(c\) has vertical differential zero, so its total differential is also \(d_hc\). The original total class is zero, and (S.5) makes its global class zero. For \(n=1\) there are no intermediate positive vertical degrees, so the same argument starts the induction. This proves (S.6). It is the finite-cover criterion of Cohomology on sites, Theorem 4.1, here with its finite total-complex calculation made explicit; compare [Stacks, finite-cover acyclicity].

S.4. Pointwise contractions preserve finite support

Apply this criterion to \(F=\mathscr D_A\). Fix a finite cover \(\mathcal V\) of a basis object \(U\). For each closed point \(y\in U\), let

This is a nonempty finite set: the cover is surjective and étale, and the residue field of \(y\) is algebraically closed. Choose one element \(e_y\in E_y\). These choices are set-theoretic; they need not form a section of the scheme covering.

Using (S.2) on every intersection, finite products commute with the point sums and give

Indeed the points in a fibre of \(V_{i_0}\times_U\cdots\times_U V_{i_p}\) are precisely the tuples of points in those chosen fibres over the same \(y\). The alternating restriction differential becomes omission of the entries of a tuple, independently for each \(y\). The augmentation is the constant-function map from \(\bigoplus_yA_{f(y)}\).

For \(p\geq1\) define

and in degree zero put \((hc)_y=c_y(e_y)\) in the augmented term. If \(c\) is supported on finitely many \(y\)'s, so is \(hc\), with no new base point introduced. Thus (S.9) is an actual operator on the direct sum (S.8), not merely on the larger product over all points.

Expanding the omission differential proves \(d_hh+hd_h=1\): the term deleting the inserted \(e_y\) is \(c_y\), and every other term has an equal term of opposite sign in the other composition. In augmented degree \(-1\), evaluation of a constant function is the original coefficient, so the identity holds there too. The augmented Čech complex is contracted. Therefore its positive Čech groups vanish for every finite basis cover. Applying S.3 gives

Taking \(U=X\) and every \(A_x=\mathbb Z\) proves the divisor-sheaf assertion in R.5. This proof does not claim that all infinite sums are acyclic or that Čech cochains on one fixed cover compute arbitrary sheaf cohomology. The essential extra fact is that the support-preserving contraction works on every finite basis cover; S.3 then supplies precisely the needed derived conclusion.

Appendix T. From local lifts to the cohomology exact sequence

The curve argument uses cohomology of abelian sheaves, including the non-torsion sheaves of rational units and divisors. We supply the ordinary resolution foundations at that scope. Work on a small site with fibre products and covering families, using the axiom of choice. For the curve, a small basis as in S.1 suffices; the equivalence with the full small étale site is proved below. These constructions concern resolutions of sheaves in nonnegative degrees.

T.1. Local representatives, exactness and the choice of basis

For an abelian presheaf \(M\), let \(M^+(U)\) consist of matching families of sections on covers of \(U\), identifying two families when they agree on a common refinement. Such a refinement exists by taking pairwise fibre products. Two choices of refinement arrows have the same effect on a matching family: the two arrows from one member factor together through an overlap, where the matching equality applies. Taking a further common refinement proves transitivity of the equivalence relation. Addition is componentwise after taking a common refinement. Pulling back a cover defines restrictions, and the associativity of fibre products proves that restrictions compose. The identity cover defines \(M\to M^+\).

A class represented by \((m_i)\) becomes the image of \(m_i\) on its covering member. Two original sections have the same image precisely when they agree locally. These facts also show that \(M^+\) is separated: if two classes agree on a cover, refine their representatives to that cover, then refine once more to witness each equality in \(M\). Composing covers witnesses equality of the original classes.

If \(M\) is separated, \(M\to M^+\) is injective on sections. Given a matching family of plus-classes, represent each locally in \(M\). On every overlap their images in \(M^+\) agree; injectivity makes the representatives themselves agree there. The composite cover therefore carries one matching family in \(M\). Its class glues the original plus-classes, and separation proves uniqueness. Thus

is a sheaf. Every section of \(aM\) is locally represented in \(M\), and equality of two such representatives is witnessed locally in \(M\). A map from \(M\) to a sheaf extends by gluing matching families twice. Local representation and separation make the extension unique. This proves the associated-sheaf universal property, including its naturality.

Here are its exactness consequences. For a presheaf map \(u:M\to N\), a section of \(aM\) in the kernel of \(au\) can locally be represented by \(m\in M\); its image is locally zero in \(N\). After refining, the representative lies in \(\ker u\). Local equality proves injectivity of the resulting kernel comparison. For the cokernel, representatives are locally classes in \(N/uM\); its universal property follows by factoring maps which kill \(uM\). Consequently

Kernels of sheaf maps are computed on sections, since compatible kernel sections glue to a section still killed by the map. Cokernels are sheafified presheaf cokernels. Sectionwise, the map from the quotient by the kernel to the image sends the class of \(m\) to \(u(m)\); it is well-defined, injective and surjective by the definitions. Sheafifying this isomorphism proves the coimage-to-image axiom. Finite sums are biproducts, so these constructions form an abelian category. A sheaf map is an epimorphism exactly when every target section lifts on a cover, by the local-zero description of its cokernel. It need not be surjective on sections over the original object. A monomorphism is sectionwise injective, by the kernel description.

Arbitrary sums and colimits are obtained by sheafifying their presheaf versions: a map from that associated sheaf is exactly a compatible family of maps from the summands or diagram objects. Filtered colimits are exact. At the presheaf level, a representative and any finite equation involving representatives occur at one common later index; an equation which becomes zero is zero at some later index. This proves preservation of kernels and finite sums for filtered diagrams of abelian groups. Apply this objectwise. For a diagram exact as sheaves, each remaining failure of presheaf surjectivity is locally zero; any element of its filtered colimit comes from one such stage and is still locally zero. Applying (T.2) proves exactness in sheaves. The same finite-support argument proves that a sum of monomorphisms is monic. In particular, an increasing union of subobjects is a subobject, and filtered unions commute with pullbacks, since a pullback is the kernel of a map from a finite direct sum.

To justify using the basis \(\mathcal B\) of S.1, restrict a sheaf to its basis objects. In the other direction, for a basis sheaf \(F\), assign to an arbitrary étale object \(U\) the group of families

compatible under arrows between basis objects over \(U\). If \(U\) itself is a basis object, evaluation at its identity and restriction from that value are inverse, so (T.3) gives \(F(U)\). It is a sheaf on the full site. Indeed, for a matching family on a cover of \(U\), pull the cover back to any \(W\to U\), refine by basis objects, and glue in \(F\). A common refinement proves independence of the choices, compatibility with further arrows, and uniqueness. On basis objects the chosen refinements may be reduced to finite covers by S.1. Every full-site sheaf is recovered in this manner: cover \(U\) by basis objects and glue its compatible sections; overlaps can again be covered by basis objects. These two constructions are inverse on maps as well as objects. The local-lifting criterion proves their exactness. An exact equivalence preserves injectives by transporting the defining extension problem, and preserves resolutions and sections on \(X\). Thus it preserves the cohomology used here. The general sheafification construction is also proved in Sites and sheaves, Sections 2–4; compare [Stacks, abelian sheaf foundations].

T.2. An injective target for every abelian sheaf

For each basis object \(U\), sheafify the free abelian presheaf on arrows into \(U\), and denote the result by \(P_U\). A map \(P_U\to F\) is determined by the image of the identity arrow. Conversely a section \(s\in F(U)\) sends a generator \([b:V\to U]\) to \(b^*s\). Linearity, restriction and (T.1) give inverse constructions. Hence

The object \(P\) detects nonzero maps: a section on which a map is nonzero gives a map from one summand of \(P\); set all other summands to zero. Subobjects of \(P\) form a set, since they are specified by compatible subgroups of its section groups on a small set of objects.

We first prove an extension test. Suppose every map \(K\to J\), for every subobject \(K\subset P\), extends to \(P\). For an arbitrary inclusion \(A\subset B\) and map \(A\to J\), order its extensions to intermediate subobjects of \(B\) by extension. A chain has a union by T.1, with a compatible map to \(J\). Zorn's lemma supplies a maximal extension \(C\to J\). If \(C\ne B\), (T.4) detects the nonzero quotient map: there is \(h:P\to B\) with image not contained in \(C\). Put \(K=h^{-1}C\), and extend the induced \(K\to J\) to \(v:P\to J\). The map \(C\oplus P\to J\) which adds the old map and \(v\) kills every pair \((-h(k),k)\). Those pairs are exactly the kernel of \((c,p)\mapsto c+h(p)\). It descends to \(C+\operatorname{im}h\), contradicting maximality. Thus the extension exists on \(B\), which says that \(J\) is injective.

For a sheaf \(M\), take all pairs \((K,v)\) with \(K\subset P\) and \(v:K\to M\), and form

The quotient is a sheaf quotient. The map \(M\to T(M)\) is monic: locally, a relation whose \(P\)-coordinates vanish has every \(k\)-coordinate zero, because each \(K\to P\) is monic; its \(M\)-coordinate is therefore zero too. For every specified pair the corresponding copy of \(P\) supplies the desired extension into \(T(M)\).

Starting with \(M_0=M\), use (T.5) at successor stages and filtered union at limit stages. All maps remain monic. We recall the cardinal bound used to stop this construction. For every infinite cardinal \(\kappa\), \(|\kappa\times\kappa|=\kappa\). Induct over infinite cardinals and order pairs first by their maximum, then lexicographically within that maximum. At \(\kappa=\aleph_0\) each initial segment is finite. At a later cardinal, an initial segment lies inside \((\gamma+1)^2\) for some \(\gamma<\kappa\), so by the induction hypothesis it has cardinality less than \(\kappa\). The ordering therefore has order type at most \(\kappa\): a larger order type would have an initial segment of order type \(\kappa\). The diagonal gives the reverse inequality. This proves the bound.

Now take an infinite \(\kappa\) at least the size of the set of subobjects of \(P\), and let \(\beta\) be the least ordinal of cardinality greater than \(\kappa\). It is a limit ordinal, since adding one point to an infinite well-ordered set does not increase its cardinality, as also follows from the preceding bound. Its cofinality is greater than \(\kappa\): otherwise it would be the union of at most \(\kappa\) smaller ordinals, each of cardinality at most \(\kappa\), hence itself of cardinality at most \(\kappa\), a contradiction.

Given \(v:K\to M_\beta\), form the increasing subobjects \(K_\alpha=v^{-1}M_\alpha\). Their union is \(K\), by the pullback assertion in T.1. There are at most \(\kappa\) distinct such subobjects. Choose an index for each; the cofinality bound puts all these indices below one \(\alpha<\beta\). Thus \(K_\alpha=K\), so \(v\) factors through \(M_\alpha\). At the next stage it extends to \(P\), with values in \(M_\beta\). The extension test proves that \(M_\beta\) is injective. We have proved enough injectives without assuming that the site has enough points. The programme construction is Cohomology on sites, Proposition 1.1.

T.3. Maps and homotopies between ordinary resolutions

Embed \(F\) into an injective \(I^0\), embed its cokernel into \(I^1\), and continue. The quotient followed by the next embedding defines the differential. Its kernel is the preceding image, giving an exact augmented complex \(0\to F\to I^\bullet\).

A map \(f:F\to G\) extends to a chain map between any chosen injective resolutions \(I^\bullet,J^\bullet\). Extend first from \(F\subset I^0\) into \(J^0\). Having defined the map through degree \(n\), the composite \(d_J^nf^n\) kills \(\ker d_I^n\): in degree zero this kernel is \(F\), and later it is \(\operatorname{im}d_I^{n-1}\), where the assertion follows from the previous chain-map identity. It therefore defines a map on \(\operatorname{im}d_I^n\subset I^{n+1}\). Injectivity of \(J^{n+1}\) extends it to \(f^{n+1}\). This proves the induction.

If \(f^\bullet,g^\bullet\) extend the same sheaf map, write \(w=f-g\). Construct maps \(h^{n+1}:I^{n+1}\to J^n\) so that

For \(n=0\), \(w^0\) vanishes on \(F\), so it factors through \(\operatorname{im}d_I^0\); injectivity of \(J^0\) extends this factor to \(h^1\). Inductively, set \(r^n=w^n-d_J^{n-1}h^n\). On the preceding image, the chain-map identity and the already proved equation give

Thus \(r^n\) factors through \(\operatorname{im}d_I^n\), and injectivity of \(J^n\) supplies \(h^{n+1}\). This proves (T.6) in every degree. Applying sections preserves this homotopy equation. It follows that

has well-defined functorial maps. Comparisons in both directions between two resolutions have composites homotopic to the identities, since those composites extend the identity of \(F\). Hence (T.8) is independent of the resolution. Its degree-zero group is \(F(U)\). This argument does not assert that a termwise-injective unbounded complex has the same properties.

T.4. Constructing the connecting map and checking exactness

Consider \(0\to F\to G\xrightarrow{\pi}H\to0\). Choose embeddings \(F\to I_F^0\) and \(H\to I_H^0\) into injectives. Extend the former across \(G\), giving \(u:G\to I_F^0\). Then

is monic: zero in its second coordinate puts \(g\) in \(F\), and its first coordinate then detects \(g\). The induced quotient sequence

is exact. To check its middle kernel, work with local representatives. If the second coordinate lies in \(H\), lift that section locally to \(G\) and subtract (T.9); the remaining representative has second coordinate zero. A representative \((a,0)\) is in the image of \(G\) exactly when it comes from \(F\), proving injectivity on the left. The right quotient is locally surjective. T.1 turns these checks into exactness of sheaves.

Repeat the same construction for (T.10), and for each subsequent quotient sequence. Finite sums of injectives are injective, because extension can be done in the two coordinates. Quotient followed by the next embedding gives compatible differentials and an exact sequence of resolutions

Every degree sequence is split as sheaves, so applying sections over \(U\) leaves it exact. The splittings need not commute with the differentials.

For a cycle \(c\in I_H^n(U)\), lift it to \(b\in I_G^n(U)\). Since \(dc=0\), there is a unique \(a\in I_F^{n+1}(U)\) whose image is \(db\). It is a cycle because \(d^2b=0\) and the left inclusion is injective. Define

Changing \(b\) by a left section changes \(a\) by its differential. Changing \(c\) by \(de\) permits changing \(b\) by the differential of a lift of \(e\), which does not change \(a\). Thus (T.12) is well-defined. It is additive by adding lifts.

These maps yield an exact sequence

Here are the three checks in every degree. If a middle cycle maps to a boundary on the right, subtract the differential of a lift of that boundary's primitive; the result is a cycle from the left. If \(\delta[c]=0\), write \(a=da_0\); then \(b-a_0\) is a middle cycle lifting \(c\). If a left cycle becomes a middle boundary \(db\), the projection of \(b\) is a right cycle and its connecting class is that left class. Conversely the same formulas show that successive maps have zero composite. The first three terms are ordinary left exactness of sections from T.1. This proves (T.13) without an assumed long exact sequence theorem.

For completeness, compatible comparisons can be chosen for any map between two original short exact sequences. At the initial stage choose extensions on \(I_F^0\) and \(I_H^0\) into the corresponding target injectives \(J_F^0,J_H^0\). The required middle map has block form

On \(G\), the discrepancy between the prescribed first coordinate and \(fu\) vanishes on \(F\); it therefore factors through \(H\). Extend that map from \(H\subset I_H^0\) into the injective \(J_F^0\) to obtain \(t\). The second coordinate already agrees by the chosen extension on \(H\). Thus (T.14) gives a commuting diagram, which induces a map of the quotient sequences (T.10). Repeat at each stage. The resulting comparison is a map of the entire diagram (T.11). Applying it to a lift in (T.12) proves naturality. T.3 shows that all its component maps induce the unique cohomology maps; applying this to the identity original sequence also proves independence of all resolution and extension choices. Restrictions of sections commute with the construction by the same lift formula.

Together, T.1–T.4 and the exact slice adjoint proved in S.2 provide the abelian-sheaf foundations for S.3 and for the long exact divisor sequence used in R.5. They do not establish the additional geometric, unbounded or normalized adic results required elsewhere in the lesson.

Appendix U. Finite cohomology models and divisors in families

Retain the fixed smooth connected projective curve \(C/k\) of Appendix Q, with \(k\) algebraically closed. We prove the two algebraic assertions used there: the two-term cohomology model survives every change of parameter ring, and finite divisors and fibrewise nonzero sections give relative Cartier divisors over arbitrary parameter schemes.

U.1. Exact tensoring and a finite replacement

We first record an elementary tensor argument. If

is exact and \(Q\) is flat over a ring \(A\), the sequence remains exact after tensoring with any module \(M\). Here is the assertion at its only nonautomatic position. Choose a free presentation \(V\to F\to M\to0\), where \(V\) is the kernel in the free module \(F\). Lift an element of \(E\otimes M\) that dies in \(D\otimes M\) to \(e\in E\otimes F\). Its image comes from some \(v\in D\otimes V\). The image of \(v\) in \(Q\otimes V\) dies in \(Q\otimes F\), hence is zero by flatness. Right exactness lifts \(v\) from \(E\otimes V\). Subtract this lift from \(e\); the difference dies in \(D\otimes F\), and is zero since \(F\) is free. Thus the original element is zero. If \(D\) is flat too, then \(E\) is flat: for an injection \(M\to M'\), embed \(E\otimes M\) in \(D\otimes M\) and use the injection into \(D\otimes M'\).

Now let \(A\) be Noetherian and let

be a complex of flat modules with finite \(H^0\) and \(H^1\). The terms need not be finite. Choose a finite free module \(P\) surjecting onto \(H^1\), and lift its basis to a map \(g:P\to C^1\). The map \((d,-g):C^0\oplus P\to C^1\) is surjective. Put

The preceding argument makes \(E\) flat and makes its defining sequence universally exact. Projection to \(P\) also gives

The right module is finite because \(A\) is Noetherian, so \(E\) is finite. It is locally finite free. Explicitly, over a localization of \(A\) at a prime, lift a basis of its residue-vector-space quotient. Nakayama gives a surjection from a finite free module onto \(E\). Its kernel is finite, and flatness of \(E\) makes reduction of that sequence exact at the left. The reduced map is an isomorphism, so Nakayama kills the kernel. The resulting basis spreads to an open neighbourhood: clear its denominators, then kill the finite kernel and cokernel by elements outside the prime. For this use of Nakayama, if a finite module \(N\) satisfies \(N=IN\), express a generating column as \(B\) times that column with entries of \(B\) in \(I\). The adjugate identity shows that \(\det(1-B)\in1+I\) annihilates \(N\). It is a unit when \(I\) lies in the maximal ideal.

The maps \(E\to C^0\) and \(g:P\to C^1\) define a comparison \(K\to C\). For every \(A\)-module \(M\), universal exactness identifies \(E\otimes M\) with the kernel of

The kernel of its projection to \(P\otimes M\) is exactly \(\ker(d\otimes M)\). Its projection image is exactly the kernel of the surjection \(P\otimes M\to\operatorname{coker}(d\otimes M)\). Consequently \(K\otimes M\to C\otimes M\) induces isomorphisms in both degrees. This proves the finite replacement and its arbitrary-tensor compatibility directly. It is the two-term instance of Base change and the Grothendieck complex, Lemma 3.1.

Apply this construction to Q.3's two-affine Čech complex over a finitely generated parameter algebra. Its flat terms and finite cohomology were established there. After localizing the parameter algebra, \(E\) and \(P\) are finite free. The comparison just proved can then be tensored first to the original parameter ring and subsequently to any algebra over it. This proves precisely (Q.8), including nonflat changes of base.

U.2. A nonzerodivisor on the closed fibre

Let \((A,\mathfrak m)\to(B,\mathfrak n)\) be a flat local map of Noetherian local rings. Suppose that multiplication by \(f\in B\) is injective on \(B/\mathfrak mB\). Then

Flatness identifies each layer of the \(\mathfrak m\)-adic filtration with

It is a direct sum of copies of \(B/\mathfrak mB\), on which \(f\) is injective. If \(fz=0\), induction through these layers puts \(z\) in every \(\mathfrak m^rB\). Their intersection is zero by L.1: apply its local separation theorem to the finite \(B\)-module \(B\) and the ideal \(I=\mathfrak mB\subset\mathfrak n\). That theorem explicitly permits any ideal inside the maximal ideal. Its Artin–Rees and determinant proof is given there, so no equality \(\mathfrak mB=\mathfrak n\) is required.

For any proper ideal \(J\subset A\), the map \(A/J\to B/JB\) is again flat, local and Noetherian, with the same closed fibre. The preceding argument therefore makes \(f\) injective on \(B/JB\). Equivalently,

Since \(B\) is \(A\)-flat, \(J\otimes_A B=JB\). Right exactness identifies \(J\otimes_A(B/fB)\) with \(JB/fJB\). Equation (U.8) says that its map into \(B/fB\) is injective. The ideal criterion now proves flatness. Its proof is Tor and flat modules, Theorem 2.1: induction on the rank of a free module extends the ideal injection test to its submodules; a tensor uses finitely many coordinates, so this extends to arbitrary free modules; presenting any inclusion as quotients of two nested submodules of a free module proves preservation of every injection. This argument uses the ideal criterion directly, without a derived replacement theorem.

Thus (U.6) has been proved from the stated Noetherian and tensor facts. It is the local algebra assertion of [Stacks, Tag 00MF]; the next two parts give the fixed-curve applications of [Stacks, Tag 062Y] with their descent steps specified.

U.3. A finite flat family has an invertible ideal

First suppose that the parameter algebra \(A\) is Noetherian and \(D\subset C_A\) is finite locally free over \(A\). At a point of \(C_A\), localize both the base and the curve, writing the quotient sequence as

Both \(B\) and \(Q\) are flat over the local base: the curve is smooth, and localizing the finite flat family preserves base flatness. U.1 makes \(I\) flat too, and identifies its reduction with the ideal of the fibre subscheme. X.1 proves the discrete valuation description at a closed point from the specified smooth regularity input, over arbitrary fields; the generic local ring is a field. An ideal cutting out a finite subscheme is therefore generated by a nonzerodivisor, including the unit ideal away from the subscheme. Lift such a generator to \(f\in I\). The finite \(B\)-module \(I/Bf\) has zero reduction modulo the base maximal ideal, so Nakayama makes it zero. The resulting surjection \(B\to I\) has finite kernel. Since \(I\) is flat over the base, U.1 makes its reduction exact; the fibre map is an isomorphism onto the fibre ideal. The kernel consequently has zero reduction, and Nakayama kills it as well. Thus \(I=Bf\simeq B\), with \(f\) a nonzerodivisor. This proves the Cartier assertion at every point.

Here is the descent needed when \(A\) is arbitrary. Work locally on \(\operatorname{Spec}A\) so that the finite algebra of \(D\) is free of rank \(d\) and the support of \(D\) lies in a fixed affine open \(U\subset C\). Such opens cover the base: choose \(d+1\) distinct points of \(C(k)\), use Q.1's affine complements, and note that a fibre of length \(d\) misses at least one of those points. The locus where the family misses that point is open, since its intersection with the point's section is closed in the base.

Write \(\mathcal O(U)=k[z_1,\ldots,z_m]/(r_1,\ldots,r_s)\). The quotient to the free algebra of \(D\) is specified by commuting multiplication matrices \(Z_i\), a vector \(v\) for the unit, and finitely many polynomials \(b_j\) such that \(b_j(Z)v\) are the chosen free basis. The equations \(r_a(Z)=0\), commutativity, and these basis identities are finite. Put their coefficients into a finitely generated \(k\)-subalgebra \(A_0\subset A\). Over \(A_0\) they define a cyclic module \(Q_0=A_0^d\) over \(\mathcal O(U)\otimes_k A_0\), with cyclic vector \(v\). The kernel of evaluation at \(v\) is an ideal: if an element kills \(v\), commutativity and cyclicity make it kill the whole module. Thus \(Q_0\) is a quotient algebra, free over \(A_0\), and its base change recovers \(D\).

The Noetherian argument makes the ideal of this quotient invertible and generated locally by a nonzerodivisor. Its quotient is \(A_0\)-flat, so U.1 preserves the ideal sequence after tensoring to \(A\). Invertibility is preserved by base change, and the injection remains an injection. We obtain a relative effective Cartier divisor over \(A\). This proves Q.2's assertion for arbitrary parameter schemes, since both invertibility and the displayed local equations can be checked on affine base opens.

U.4. A fibrewise nonzero section cuts out a flat divisor

Let \(L\) be a line bundle on \(C_A\), with \(A\) any \(k\)-algebra, and let \(s\) restrict to a nonzero section on every geometric curve fibre. We prove that its zero scheme is a relative effective Cartier divisor, finite locally free over the base. Its degree on a fibre is \(\deg L_t\).

Descend \(L\) and \(s\) to a finitely generated \(k\)-subalgebra \(A_0\subset A\). Q.3 explains the finite presentations, inverse trivializations and gluing identities for the bundle. Include the finitely many coefficients of the section on a finite trivializing affine cover and its overlap identities. This gives the section as well. It remains to arrange that the descended section is nonzero on every fibre near the image of \(\operatorname{Spec}A\); that assertion cannot be inferred merely from the existence of a descended bundle.

Use U.1's finite model locally on \(\operatorname{Spec}A_0\). The descended section is a cycle \(v\in K^0\), and the same is true after every change of base. Since \(H^0(K)=\ker(K^0\to K^1)\) embeds in \(K^0\), its fibre section is zero exactly when all coordinates of \(v\) vanish in the residue field. Nonvanishing therefore defines an open subset of the base. This open contains the image of \(\operatorname{Spec}A\): extension of residue fields is faithful on vector spaces. Replace the descended base by this open and work on its affine cover.

On the Noetherian descended curve, trivialize \(L\) at a point and write \(s=f\). Each fibre is geometrically integral, so a nonzero section of a line bundle is a nonzerodivisor in every fibre local ring. U.2 shows that \(f\) is a nonzerodivisor on the curve and that its quotient is flat over the base. The zero scheme \(D_0\) is thus a relative effective Cartier divisor. U.1 shows that its defining sequence remains exact after every base change. In particular its pullback to \(A\) is exactly the zero scheme of \(s\), with invertible ideal and flat quotient.

For completeness, finiteness can be checked without an extra compactification argument. At any point of the descended base, the finite zero divisor on the fibre misses some \(q\in C(k)\); there are infinitely many such points by Q.1. Restrict to the open base locus where the section does not vanish at \(q\). Then \(D_0\) lies in \((C\setminus\{q\})\times\operatorname{Spec}A_0\), an affine scheme by Q.1, and is closed there. Its coordinate algebra is finite over \(A_0\): it is the global sections of the coherent quotient \(\mathcal O_{D_0}\) on the projective curve, so W.4's projective cohomology finiteness applies. Hence \(D_0\) is finite. A finite flat module over the Noetherian base is locally free by U.1's basis argument. These statements survive the arbitrary base change to \(A\). Finally the local discrete valuation lengths identify its fibre length with the degree of the zero divisor of \(s_t\), which equals \(\deg L_t\) by (Q.2).

This also justifies the sum of graph divisors in Q.2. A graph is a closed finite locally free family of degree one, so U.3 makes its ideal invertible. The product of the \(d\) graph ideals is the image of a section of their inverse tensor product. On each curve fibre its equation is a product of parameters, with repetitions allowed, and is nonzero. The argument above makes its zero scheme finite locally free of degree \(d\). In Q.4 apply the same result to the evaluation section of \(L\otimes f^*(f_*L)^{-1}\). Its fibre is the unique nonzero section up to scalar, and its degree is \(g\). Thus the divisor construction and its inverse in (Q.10) hold over all parameter schemes and commute with arbitrary base change.

Appendix V. Coordinates on the finite-divisor scheme

Let \(C/k\) and \(H_d\) be as in Q.2. We construct étale charts \(H_d\supset V\to\mathbf A^d_k\) around every closed point. This proves smoothness directly and also identifies the completed local rings in (Q.7).

V.1. One coordinate at all support points

Write a nonempty divisor as \(D=\sum_{i=1}^r d_i x_i\), with distinct \(x_i\in C(k)\). Q.1 supplies an affine open \(U_0\) containing the support: remove a different rational point of \(C\). Let \(B=\mathcal O(U_0)\) and let \(\mathfrak m_i\) be the maximal ideal of \(x_i\). Choose \(u_i\in\mathfrak m_i\) nonzero in \(\mathfrak m_i/\mathfrak m_i^2\), which is one-dimensional because \(C\) is smooth of dimension one. Choose distinct \(a_i\in k\). The Chinese remainder theorem gives \(t\in B\) satisfying

The ideals \(\mathfrak m_i^2\) are pairwise comaximal. For example, if \(a+b=1\) with \(a\in\mathfrak m_i\), \(b\in\mathfrak m_j\), expanding \((a+b)^3\) puts \(1\) in their sum of squares. For two comaximal ideals \(I,J\), a decomposition \(1=e+f\), \(e\in I,f\in J\), makes \(fy+ez\) a simultaneous representative of the residues \(y\bmod I,z\bmod J\); induction proves the finite version used in (V.1).

Shrink to an affine neighbourhood \(U\) of all \(x_i\) where \(dt\) is a basis of \(\Omega_{C/k}\). To obtain one affine neighbourhood, the complement of this locus in \(U_0\) is closed and misses all \(x_i\). Its ideal contains an element equal to one at each \(x_i\), again by the Chinese remainder calculation. Invert that element. The map

is étale. Here is a lifting proof that also fixes the precise coordinate criterion. Work on a smooth affine chart of \(U\), with \(dt\) a differential basis. Given a square-zero extension \(A'\twoheadrightarrow A\), ideal \(J\), and a compatible map to the chart over \(k[T]\), first lift it as a \(k\)-map using smoothness. If \(b\in A'\) is the prescribed image of \(T\), its discrepancy \(e=\widetilde t-b\) lies in \(J\). The universal property of differentials gives a unique derivation into \(J\) sending \(dt\) to \(e\). Subtract it from the lift. Because \(J^2=0\), the product rule says that the corrected map is a ring map, now with \(t\mapsto b\). Two such maps differ by a derivation vanishing on \(dt\), hence are equal. This proves formal étaleness on these charts. The lifts agree on overlaps by uniqueness, so the assertion holds for (V.2). Finite presentation over \(k[T]\) follows by adjoining the single equation identifying \(T\) with the chosen polynomial representative of \(t\).

The algebraic prerequisites for this argument have explicit proofs in Formally smooth, unramified and étale ring maps, Sections 1–5: square-zero differences are derivations; a smooth presentation has a split conormal sequence; choosing independent rows gives standard smooth neighbourhoods; and an invertible Jacobian solves the lifting equations. The differential universal property, quotient and localization formulas are proved in Kähler differentials, Sections 1–3 and 5. The rank-one assertion for a smooth curve is the standard-chart calculation in Smooth morphisms, Proposition 1.1.

V.2. A divisor recorded by one monic polynomial

The image of \(t-a_i\) is a uniformizer at \(x_i\). In the quotient of its discrete valuation ring by \((t-a_i)^{d_i}\), the powers from zero to \(d_i-1\) are a basis over \(k\): reduce the constant coefficient modulo the maximal ideal and iterate through its successive principal powers. Independence follows by taking the least nonzero coefficient and its valuation. Since the \(a_i\) are distinct, Chinese remainders now give

In particular \(1,t,\ldots,t^{d-1}\) form a basis of the length-\(d\) algebra.

Inside the open scheme \(H_d(U)\subset H_d(C)\), let \(V_t\) be the locus where these powers form a basis of the universal finite algebra. It is open: in any local basis the condition is the invertibility of a determinant. It contains \(D\). The unique relation expressing \(t^d\) in that basis has the form

Its coefficients are regular functions on \(V_t\), since the inverse basis matrix is regular where its determinant is invertible. They define

For any parameter ring \(A\), the algebra of a divisor in this chart is canonically \(A[T]/(F)\), with \(T\) acting as \(t\). Monic division makes the source free on the same \(d\) powers, and the map to the divisor algebra carries that basis to a basis, proving the assertion over arbitrary \(A\). Thus a chart point consists of its monic polynomial together with its embedded map into \(U\) over the coordinate line.

The construction of \(H_d(U)\) used here is the finite commuting-matrix construction in Q.2, with full proof in Hilbert and Quot schemes, Proposition 5.1. The openness of \(H_d(U)\) follows because a finite family is contained in \(U\) exactly off the closed image of its intersection with \(C\setminus U\). No projectivity theorem for a general Hilbert functor enters this chart.

V.3. Unique lifting proves that the chart is étale

Take any square-zero extension \(A'\twoheadrightarrow A\), a divisor \(Z\) in \(V_t(A)\), and a lift \(F'\in A'[T]\) of its polynomial (V.4), with the same monic leading coefficient. Set

It is free of rank \(d\) over \(A'\), and its reduction is canonically \(Z\). The map \(Z\to U\) is over \(\mathbf A^1\). Formal étaleness of (V.2) lifts it uniquely to \(Z'\to U\). The lift respects \(A'\), so gives \(Z'\to U_{A'}\). Local lifting and uniqueness glue on \(Z'\); only a square-zero ideal is being tested.

This map is a closed immersion. On rings its map \(\mathcal O(U)\otimes_k A'\to Q'\) sends the coordinate \(t\) to the class of \(T\). That class together with \(A'\) generates the whole quotient algebra, so the map is surjective. The powers of \(t\) still form the monic basis, making this a point of \(V_t(A')\). Every candidate lift has the algebra (V.6) by V.2, and its map to \(U\) is the unique étale lift just constructed. Thus it is unique. We have proved every square-zero lifting test for \(\Phi_t\).

The map \(\Phi_t\) is locally of finite presentation: its source is locally of finite type over the Noetherian field \(k\), and an affine presentation over \(k[c_1,\ldots,c_d]\) adds finitely many equations for these coefficient functions. Finite presentation and the proved formal lifting property make it étale. Equivalently, the local algebra criterion is the definition and standard-form theorem in the ring-map lesson linked in V.1. On an affine open of \(V_t\), the lift lands in that same open because square-zero thickenings preserve the underlying points.

It follows that \(V_t\) is smooth over \(k\): lift the \(d\) coefficients freely, then use the unique lift over them. Moreover its differentials over \(k\) are freely generated by \(dc_1,\ldots,dc_d\). To see this without just using a rank formula, the unique-lift test with split square-zero rings says that every derivation on the polynomial coefficient ring into any module extends uniquely to the chart ring. The representing property of differentials makes their natural map an isomorphism. Y.5–Y.6 prove polynomial local regularity and the standard smooth presentation calculation. Applied here, (Y.16) gives regular closed local rings of dimension \(d\); thus the chart is smooth of dimension \(d\).

Every closed point of \(H_d\) has the chart just constructed, since \(k\) is algebraically closed. These opens cover \(H_d\): their closed complement in a finite-type \(k\)-scheme, if nonempty, would contain a closed point by the weak Nullstellensatz. For \(d=0\), \(H_0=\operatorname{Spec}k\) already has the required property. Thus \(H_d\) is smooth of dimension \(d\) in every characteristic.

V.4. Completed coefficients and separate support clusters

Let \(D\) correspond to coefficients \(b_1,\ldots,b_d\in k\) in (V.4). Put \(A=\mathcal O_{H_d,D}\). The chart gives a continuous map

We construct its inverse explicitly. Over each Artinian ring \(R_n=k[X_1,\ldots,X_d]/(X_1,\ldots,X_d)^n\), the coefficient tuple \((b_j+X_j)_j\) has a unique lift to \(V_t\) reducing to \(D\). Apply V.3 successively to the square-zero kernels of \(R_{n+1}\to R_n\). Uniqueness makes the resulting maps \(A\to R_n\) compatible. They give a local map \(A\to k[[X_1,\ldots,X_d]]\), continuous for the maximal-ideal topologies, and hence an extension \(\beta:\widehat A\to k[[X_1,\ldots,X_d]]\). Indeed its map to \(R_n\) kills \(\mathfrak m_A^n\), so it factors through that quotient and is defined on the inverse limit.

The composite \(\beta\alpha\) fixes every \(X_j\), hence every power series by continuity. For \(\alpha\beta\), reduce to each \(A/\mathfrak m_A^n\). Its induced map from \(A\) and the canonical quotient map have the same coefficient tuple and the same closed point. V.3's uniqueness, iterated through the nilpotent maximal ideal, makes them equal. Equality in every quotient makes \(\alpha\beta=1\). Thus (V.7) is an isomorphism; no comparison of dimension before and after completion is needed for this conclusion.

To obtain exactly the separate coefficients in (Q.6)–(Q.7), let \(R\) be any local Artinian \(k\)-algebra with residue field \(k\). A deformation of \(D\) has a finite free algebra whose special fibre is the product of the support clusters. The idempotents lift uniquely. For a square-zero ideal this follows by correcting a lift \(a\) of an idempotent by

Here \((2a-1)^2=1+4(a^2-a)\) is a unit. Two corrected idempotents differ by an element killed by their sum minus one, which is a unit, proving uniqueness. Iterate through the powers of the nilpotent maximal ideal. Orthogonality and summing to one persist by uniqueness. This gives the individual free rank-\(d_i\) algebras.

The coordinate \(t-a_i\) identifies the completed curve ring at \(x_i\) with \(k[[t-a_i]]\), by the same étale lifting argument as (V.7), now for (V.2). Any other formal parameter \(u_i\) has nonzero linear term in this coordinate; coefficient-by-coefficient inversion gives the inverse power series. In an Artinian deformation the coordinate is nilpotent, so those series evaluate by finite sums. The powers \(1,u_i,\ldots,u_i^{d_i-1}\) reduce to a basis, hence are a basis of the free cluster algebra: Nakayama gives a surjection between free modules of the same rank, and its determinant is a unit. The unique monic relation is precisely (Q.6), with every \(a_{i,j}\in\mathfrak m_R\).

Conversely, any such coefficient list gives a free monic quotient. If \(\mathfrak m_R^N=0\), its relation implies \(u_i^{d_iN}=0\). Therefore it defines a map from the completed curve ring, and thus a map to a neighbourhood of \(x_i\). It is a closed embedded cluster: the actual coordinate \(t-a_i\) and \(u_i\) are inverse polynomial expressions on this nilpotent quotient. Distinct clusters combine to a closed divisor; the polynomials in \(t\) supported at the distinct residues \(a_i\) are comaximal, so their quotient factors separate the clusters. These constructions are inverse and commute with all Artinian base changes.

The independent coefficient lists consequently describe the completed local ring. More explicitly, lift the universal lists over the quotients of \(k[[a_{i,j}]]\) by powers of its maximal ideal to obtain a map from \(A\); in the other direction, apply the coefficient construction to every \(A/\mathfrak m_A^n\). Their inverse-limit maps are inverse because the two constructions are inverse on each Artinian quotient, exactly as above. This proves (Q.7), including repeated points and all characteristics, with the specific coefficient coordinates asserted there.

Appendix W. The coherent cohomology used by the Picard construction

Q.1 and Q.3 need a specific chain of cohomology statements: affine vanishing, a finite ordered Čech model, and finiteness on a projective scheme over a Noetherian ring. We give that chain here. Throughout this appendix cohomology is ordinary Zariski sheaf cohomology, with the abelian-sheaf resolutions of T.1–T.4. No assertion about unbounded or adic complexes is involved.

W.1. Modules, affine sections and the principal-cover contraction

For an \(A\)-module \(M\), localization is exact. A fraction vanishes exactly when an allowed denominator kills its numerator; this criterion proves injectivity and exactness in the middle, and lifting a numerator proves surjectivity. The maps \((a/s)\otimes m\mapsto am/s\) and \(m/s\mapsto (1/s)\otimes m\) identify localization with tensoring by the localized ring.

Every cover of \(\operatorname{Spec}A\) has a finite refinement by principal opens. Indeed, principal opens form its defining basis. If their defining elements generate a proper ideal, a maximal ideal containing it gives a point outside the cover. Otherwise a finite expression for \(1\) supplies the finite refinement. The same reasoning in \(A_f\) applies to \(D(f)\). Thus these basis objects are quasi-compact and their finite intersections are again basis objects.

Take a finite principal cover \(D(f_1),\ldots,D(f_r)\). The augmented ordered complex is

To prove exactness, localize at one \(f_j\). That member of the cover is now the whole space. Extend an ordered cochain to alternating tuples, with zero on repetitions, and define \((hc)_{i_0\ldots i_{p-1}}=c_{j,i_0,\ldots,i_{p-1}}\); the degree-zero map is evaluation at \(j\) into the augmentation. The alternating differential satisfies \(dh+hd=1\): deleting the inserted index supplies the original term, and all other terms cancel in pairs. The full Čech complex with all tuples, as used in S.2, has the same contraction without the alternating extension. Localization commutes with the finite products in each degree and with cohomology. Hence every cohomology module of either augmented complex becomes zero after inverting each \(f_j\). Such a module is zero: for any element choose a common power \(f_j^a\) killing it for all \(j\); those powers still generate the unit ideal. This proves the assertion before localization.

The equalizer part of (W.1) constructs \(\widetilde M\) on the principal basis, with sections \(M_f\) and stalk \(M_{\mathfrak p}\). Finite refinements give its sheaf property on all covers. We also need the converse: if \(F\) is quasi-coherent on this affine scheme, then

Here are the gluing details. Restriction of an associated module sheaf from \(\operatorname{Spec}B\) to an affine open \(\operatorname{Spec}C\) is associated to \(C\otimes_B M\): the natural map is an isomorphism on stalks, since the local rings of an open subscheme are the same. Consequently quasi-coherence supplies a finite principal cover of \(\operatorname{Spec}A\) on which \(F\) is associated to modules \(N_i\). The sheaf equalizer expresses its global sections as the kernel of the difference map from \(\prod_i N_i\) to the product of overlap sections. Localize this finite equalizer at \(f\). Exactness of localization identifies it with the equalizer for the induced cover of \(D(f)\), proving (W.2). The map from the associated global-section module to \(F\) sends \(m/f^a\) to \(f^{-a}m\) on that open; (W.2) proves it is an isomorphism. Association is exact on stalks, so its inverse, affine sections on quasi-coherent sheaves, is exact too. Tensor products and pullbacks on affines correspond to module tensor products by the same stalk calculation. These arguments also prove that kernels of quasi-coherent maps remain quasi-coherent. Compare the complete affine construction in Quasi-coherent sheaves, Section 1.

W.2. From those contractions to derived cohomology

Apply the proof of S.3 on the Zariski principal basis just checked in W.1. Its hypotheses hold: intersections remain in the basis, covers have finite refinements, and the full Čech complexes of \(\widetilde M\) are acyclic in positive degrees on every finite basis cover. T.1–T.4 construct the required injective resolutions on this site. More explicitly, a positive-degree resolution cycle becomes a boundary on a finite cover. S.2 identifies the total Čech-resolution complex with global resolution sections. Induction on the degree then removes the positive vertical components on the intersections, until only a Čech cycle of \(F\) remains; W.1 makes that cycle a boundary. This proves

for any ring and module. The steps are precisely S.2–S.3's signed, finite-diagonal elimination, so the passage uses derived cohomology rather than merely exactness of affine sections within the quasi-coherent subcategory. See also Affine cohomology, Sections 2–3 and [Stacks, coherent cohomology prerequisites].

For use of an ordered finite cover, its augmented free sheaf chain complex is exact as well. At any stalk choose a covering index containing that point; insert it, with the sign of its position in the ordered tuple. The resulting simplex contraction is the identity up to the boundary. Applying \(\operatorname{Hom}(-,I)\) to this exact complex for an injective sheaf \(I\) gives exact augmented ordered Čech rows. Thus S.2's row elimination applies to the ordered double complex as well.

Let \(U_1,\ldots,U_r\) cover a separated scheme by affines. Each intersection is affine: for two such opens, their intersection is the inverse image of the closed diagonal in the affine product, hence a closed subscheme of an affine scheme; repeat for more opens. Apply an injective resolution of \(F\), with double differential \(d_h+(-1)^p d_v\) in column \(p\). Row elimination gives global cohomology. By (W.3), every column has cohomology only in vertical degree zero, where it is the corresponding section module of \(F\). To verify the other comparison explicitly, start with a total cycle and its lowest nonzero column having positive vertical degree. That component is a vertical cycle; choose a primitive, with sign \((-1)^p\), and subtract its total differential. This removes it and changes only the next column. Continue until the representative has vertical degree zero. For injectivity, apply the same procedure to a primitive whose total differential lies in that bottom row; positive vertical components of the primitive can successively be removed without changing its differential. Both procedures terminate on the finite diagonal. Hence the ordered section complex computes \(H^q(X,F)\), and

All maps are restriction maps and module maps over the base ring. In particular Q.1's two affine opens compute cohomology in degrees zero and one, also after any change of parameter ring: the base-changed opens and their intersection are still affine, and their associated modules tensor by (W.2).

W.3. Polynomial weights calculate every projective twist

Write \(P=\mathbf P_A^N\), with homogeneous coordinates \(T_0,\ldots,T_N\). On its standard charts the frame for \(\mathcal O_P(m)\) is \(T_i^m\). On the intersection indexed by a nonempty subset \(I\), its section module is the degree-\(m\) part of \(A[T_0,\ldots,T_N,T_i^{-1}:i\in I]\). These formulas follow directly by identifying the frames through the coordinate ratios. They also give the affine intersections needed in W.2.

Assume first \(N>0\). Decompose the ordered complex by Laurent monomial exponents \(e=(e_0,\ldots,e_N)\) with \(\sum e_i=m\). A monomial occurs at \(I\) exactly when \(I\) contains its negative support \(S(e)=\{i:e_i<0\}\). For that weight every occurring component is one copy of \(A\), and its differential is the signed identity to the larger allowed subsets. There are three cases. If the negative support is empty, this is the simplex cochain complex: augment it by \(A\) and insert any vertex to contract it, leaving just \(A\) in degree zero. If the negative support is a nonempty proper subset, insert a vertex outside it; the same cancellation contracts the complex without augmentation. In degree zero the potentially extra constant term is zero because the singleton at the inserted vertex is not allowed. If every exponent is negative, only the full subset occurs, giving \(A\) in degree \(N\).

Each Laurent polynomial has finite support; there are only finitely many subsets in each degree. The whole complex is therefore the direct sum of these weight complexes, and kernels and images in a direct sum are computed componentwise. We have proved

For fixed \(m\) each displayed sum is finite, including when it is empty. Thus every group is finite free, and positive-degree groups vanish for \(m\geq0\). If \(N=0\), there is one chart, every twist is the trivial line, and the answer instead is \(A\) in degree zero for every \(m\), zero in higher degrees. The decomposition and contractions involve only signed identity maps, so these particular cohomology calculations commute with any ring change. This assertion is about twists, not arbitrary coherent sheaves. It is the calculation in Cohomology of projective space, Sections 1–3; see also [Stacks, coherent cohomology prerequisites] for the coefficient-wise contraction.

W.4. Finite generation, descending induction and the fixed curve

Now suppose \(A\) is Noetherian. Polynomial rings over \(A\) are Noetherian: in one variable, choose finitely many polynomials in an ideal whose leading coefficients generate the ideal of all leading coefficients. Above the maximum of their degrees, subtract multiples to lower the degree. The remaining elements of smaller degree form a submodule of a finite free \(A\)-module, hence have finitely many generators. Those elements and the chosen polynomials generate the ideal. Repeat one variable at a time. Here finite free modules are Noetherian by induction on rank, and their finite quotients are Noetherian. Localizations and quotients of Noetherian rings remain Noetherian by contracting and extending ideals. In particular the standard charts of \(P\) are Noetherian, and kernels of maps of coherent sheaves are coherent by W.1.

We next construct a finite quotient by twists, rather than assuming one. Fix a section \(u\) of a coherent sheaf \(F\) on \(U_i=D_+(T_i)\). On each standard chart \(U_j\), the intersection with \(U_i\) is principal. By (W.2), \(u\) is a fraction in that chart's module. Clearing the finitely many denominators gives local sections of \(F(a)\) whose quotients by \(T_i^a\) equal \(u\) on \(U_i\). On each affine overlap the difference of these lifts vanishes after inverting \(T_i\); a power of its local equation therefore kills that difference. Choose a common power for the finitely many overlaps and multiply all lifts by it. They now glue to a section of a larger twist. The same argument kills any global twisted section that restricts to zero after this inversion. In particular every chosen local generator extends after a power, without requiring \(T_i\) to be regular on \(F\).

Choose finitely many generators on each \(U_i\) and extend them in this way. Raise their twist degrees to one common nonnegative integer \(a\), using further powers of the coordinate for their original chart. Their restrictions in that chart's frame still give the original generators. Together the resulting global sections give an exact sequence

with \(b\) finite and \(K\) coherent. For \(n\geq a\), the degree-\((n-a)\) monomials generate \(E(n)\), so their images generate \(F(n)\).

W.2 bounds cohomology by \(N\) on this cover. Descend in \(q\), proving finiteness simultaneously for all coherent sheaves. The induction starts above \(N\). The long exact sequence constructed in T.4 contains

Its left term is finite by W.3 and its right term by induction. The middle term is an extension of a quotient of the left term by a submodule of the right term. Both are finite over the Noetherian base, hence so is the middle term. This includes degree zero. For \(q>0\), apply the same descending induction to (W.6) tensored by \(\mathcal O(n)\). W.3 kills the left cohomology group for large \(n\), and induction kills the right one. Thus \(H^q(P,F(n))=0\) for large \(n\); taking the largest threshold over \(1\leq q\leq N\) proves simultaneous eventual vanishing. No finite resolution by line bundles has been assumed. These are the proofs used in Coherent sheaves on projective schemes, Sections 1–2.

If \(i:X\hookrightarrow P\) is a closed subscheme, a coherent sheaf on \(X\) gives a coherent sheaf \(i_*F\) on \(P\). On each chart this is simply its finite module viewed over a Noetherian ring mapping onto the coordinate ring of the closed subscheme. The common affine cover and W.2 give identical ordered complexes on both spaces. Their cohomology is therefore identical, proving finite coherent cohomology for every such projective \(X\).

Apply this to \(C_{A_i}\) in Q.3. The curve's fixed projective embedding remains a closed embedding after base change, so the cohomology of the descended line bundle is finite. The fixed two-affine cover remains a two-affine cover over \(A_i\). On any of its intersections the coordinate ring is \(B\otimes_k A_i\), which is \(A_i\)-flat because a \(k\)-basis of \(B\) exhibits it as a free \(A_i\)-module. An invertible module there is flat over that ring: locally its tensor operation is the identity, and exactness can be checked after localizing at all prime ideals. Composition of these exact tensor operations makes it \(A_i\)-flat. This establishes both inputs of U.1's finite replacement of the two-term complex. Q.3's finite-presentation descent followed by that replacement handles arbitrary \(A\); we make no general nonflat cohomology-base-change claim. The same projective finiteness proves that the affine zero scheme in U.4 has a finite coordinate module. AF.1 extends this finite-cohomology calculation to the finite-over-projective family in O.3; AF.2–AF.4 then supply its constant-field and connected-fibre argument after shrinking the base.

Appendix X. Valuations, rational functions and divisor degree

We supply the divisor calculations used in P.2, Q.1 and U.3. They apply after any field extension \(K/k\), where \(k\) and \(C\) are as in Q. In particular \(K\) is infinite but need not be perfect. Multiplicities below use residue-field degrees and include inseparable maps.

X.1. The local parameter and the constant field

Y.4 proves that regular local rings are domains and that independent parameter quotients are regular, using the length-growth calculation of Y.1–Y.3 and separation in L.1. Y.5 proves polynomial local regularity. Y.6 derives a standard smooth presentation directly from infinitesimal lifting and applies those results to prove smooth local regularity after every field extension. In particular the closed local rings of a smooth curve are one-dimensional regular domains. No assumption that a closed point has separable residue field is made.

On a smooth curve the local ring at a closed point has dimension one, so its maximal ideal is generated by one element \(\pi\). Indeed regularity says its cotangent space has dimension one, and Nakayama, with the determinant proof in U.1, lifts a basis to a generator of the maximal ideal. At a generic point the local ring is a field. Components of a smooth curve are disjoint: two meeting components would give two minimal primes in a local ring, contradicting the domain assertion. There are finitely many components; being disjoint closed subsets, they are also open. Thus a connected smooth curve is integral.

Here is the valuation argument for a Noetherian local domain \(R\) with nonzero maximal ideal \((\pi)\). A nonzero \(a\) cannot belong to every \(\pi^nR\). Otherwise the elements \(b_n=a/\pi^n\) lie in \(R\), and the ascending sequence of ideals \((b_n)\) stabilizes. At a stable step \(b_{n+1}=c b_n\), while \(b_n=\pi b_{n+1}\). Cancellation in the domain gives \(1=c\pi\), impossible for a nonunit \(\pi\). Choose the largest exponent \(n\) for which \(a\in\pi^nR\). The quotient \(a/\pi^n\) lies outside the maximal ideal and hence is a unit. The exponent is unique, since two different exponents would make a positive power of \(\pi\) a unit. Set \(v(a)=n\), \(v(a/b)=v(a)-v(b)\) and \(v(0)=+\infty\). Then

Every nonzero ideal is \(\pi^nR\): take the least valuation of a nonzero element in it. Its quotients have the explicit length calculation

because the successive powers have one-dimensional residue-field quotients. This proves the DVR assertion and its ideal and length statements directly from the local input. Compare Discrete valuation rings, Section 1 and [Stacks, curve divisor foundations].

The global constants and geometric integrality needed in Q can also be checked now. The integral projective curve \(C\) has a finite-dimensional \(k\)-algebra \(H^0(C,\mathcal O_C)\) by W.4. It embeds in its function field, so is a domain. Multiplication by a nonzero element is an injective endomorphism of a finite-dimensional vector space, hence surjective; the algebra is a field. As \(k\) is algebraically closed it equals \(k\). W.2's finite affine-cover complex, tensored with a field extension, gives

A nontrivial open-and-closed decomposition would give a nontrivial idempotent in this field. Thus \(C_K\) is connected. Smoothness survives field extension by base change of its standard smooth presentations; the regularity input above then makes its components disjoint and its local rings domains. Hence \(C_K\) is integral. This works for every extension \(K/k\) and proves geometric integrality as used here.

X.2. Every line bundle has a finite divisor representative

Write \(F=K(C_K)\). A rational function in \(F^\times\) has only finitely many nonzero orders. To see this, use a finite affine cover and write the function on each chart as a quotient of two nonzero elements of its coordinate domain. Their zero sets are proper closed subsets of a Noetherian integral curve, hence zero-dimensional and finite: each has finitely many irreducible components, and each component of dimension zero consists of its single closed generic point.

For a finite divisor \(D=\sum_x m_x[x]\), define \(\mathcal O(D)\) as the subsheaf of the rational-function sheaf with sections satisfying \(v_x(f)+m_x\geq0\). Near a point \(x\), choose a rational representative of a uniformizer \(\pi_x\), and discard its other finitely many zeros and poles and the other points of \(D\). On this neighbourhood the module is generated by \(\pi_x^{-m_x}\). The valuation criterion in (X.1) identifies its stalks with these free rank-one modules; equality of sheaves is checked on stalks. Thus \(\mathcal O(D)\) is invertible, including when some coefficients are negative. A rational function regular in every local ring of an open set is a section there: its local representatives agree in \(F\) and glue. This justifies the section description, not only its stalk version.

Now choose a nonzero vector \(s\) in the generic fibre of a line bundle \(L\). In each local frame write \(s=f_i e_i\). Units relating the frames have valuation zero, so the orders of the \(f_i\) agree on overlaps. The finite-cover argument above shows that they define a finite divisor \(D(s)\). Multiplication by \(s\) gives an isomorphism

At \(x\), the frame \(\pi_x^{-v_x(f_i)}\) maps to a unit times \(e_i\), verifying the assertion. Replacing \(s\) by \(as\) adds \(\operatorname{div}(a)\). Conversely a trivialization of \(\mathcal O(D)\subset F\) is multiplication by a rational generator whose orders are \(-m_x\), so \(D\) is principal. We obtain, with the indicated sign convention,

These are the constructions of The multiplicative group on a curve, Proposition 5.1, here retaining arbitrary residue fields. Tensoring the local sequence \(0\to(\pi_x)\to R\to\kappa(x)\to0\) by a line bundle proves the point sequence (Q.1). Its last term has no positive Zariski cohomology: it is quasi-coherent, and on any finite affine cover the opens containing \(x\) give a simplex complex with coefficient \(L|_x\), contracted by any such index. W.2 gives the derived assertion. Its degree-zero dimension over \(K\) is \([\kappa(x):K]\).

X.3. A nonconstant rational function gives a finite map

For \(a\in F^\times\), (X.1) says that either \(a\) or \(a^{-1}\) is regular at each point. Each condition holds on a neighbourhood of that point. The corresponding affine-coordinate maps glue to a morphism \(f_a:C_K\to\mathbf P^1_K\), because on their overlap the coordinates are reciprocal. If \(a\notin K\), this morphism is nonconstant. In fact, an element of \(F\) algebraic over \(K\) has valuation zero at every closed point when nonzero: a negative valuation contradicts its monic equation by the uniquely lowest-order leading term; apply the same argument to its inverse to exclude positive valuation. It is therefore a global regular function, and (X.3) puts it in \(K\). Thus \(a\notin K\) is transcendental and the generic point maps to the generic point of \(\mathbf P^1\).

We prove finiteness in a form also useful for degrees. Let \(f:X\to Y\) be a nonconstant morphism of integral projective curves over the infinite field \(K\), with \(Y\) separated. It is dominant: if the image of the generic point of \(X\) were a closed point, continuity would put the entire image in that point. Every closed fibre is a proper closed subset of \(X\), hence has finite support. Fix a closed point \(y\in Y\), and embed \(X\) in \(\mathbf P^N_K\). A hyperplane can be chosen to miss the fibre and not contain \(X\): for each of its finitely many closed points, vanishing is a proper linear condition on the coefficients over \(K\); include one additional closed point of \(X\) if the fibre is empty. A finite union of proper linear subspaces does not exhaust a vector space over an infinite field. For example, choose a nonzero linear form annihilating each subspace; their product is a nonzero polynomial, which cannot vanish on all \(K^{N+1}\), by induction on the number of variables.

The intersection \(Z=X\cap H\) with that hyperplane is finite, and its image in \(Y\) is a finite closed set avoiding \(y\). Images of these closed points are closed: their residue fields are finite over \(K\), so the image residue field is also algebraic and finite over \(K\). Choose an affine neighbourhood \(V\) of \(y\) disjoint from this image. The graph of \(f\) is closed in \(X\times_KY\), since \(Y\) is separated. Hence it is closed in \(\mathbf P^N_K\times_KY\), and over \(V\) it lies in

Its restriction is closed in this affine scheme, so \(f^{-1}V\) is affine. It is simultaneously a closed subscheme of \(\mathbf P^N_V\). W.4 therefore makes its coordinate module finite over the Noetherian coordinate ring of \(V\). This says precisely that \(f^{-1}V\to V\) is finite. Such neighbourhoods cover the closed points and the generic point of \(Y\), so \(f\) is finite. This proof uses projective coherent finiteness, and needs no general theorem about proper quasi-finite morphisms or formal functions. The comparison statement is [Stacks, curve divisor foundations].

X.4. Weighted fibre lengths and principal degree zero

Let \(f:X\to Y\) now be a nonconstant morphism between smooth integral projective curves over \(K\). X.3 makes it finite. At a closed point \(y\), put \(A=\mathcal O_{Y,y}\), with uniformizer \(\pi_y\). The finite semilocal algebra \(B=(f_*\mathcal O_X)_y\) is contained in \(K(X)\), and is torsion free over \(A\). Its generic fibre is the field \(K(X)\) over \(K(Y)\): inverting the nonzero elements of a target affine coordinate ring gives a finite-dimensional domain over its fraction field, hence a field with fraction field \(K(X)\). Denote its dimension by \(d\).

A finite torsion-free module \(M\) over this DVR is free. Lift a basis of \(M/\pi_yM\); Nakayama gives a surjection \(A^r\to M\). If a nonzero relation among these lifts existed, divide all its coefficients by their largest common power of \(\pi_y\). Torsion freeness ensures the divided vector is still a relation. At least one coefficient is now a unit, contradicting linear independence after reduction. Thus the lifts are a basis. Applied to \(B\), its rank is \(d\), and

This finite-dimensional algebra is the product of its local factors at the points above \(y\). One can see the decomposition directly: there are only finitely many maximal ideals, since the Chinese remainder surjections to products of their residue fields bound their number by the vector-space dimension. All primes are maximal. The nilradical is nilpotent, since it is finitely generated by nilpotent elements. The product of the distinct maximal ideals equals their intersection; a sufficiently high power is therefore zero. The Chinese remainder theorem for those powers gives the claimed local factors. In the factor at \(x\), the pullback of \(\pi_y\) is a unit times \(\pi_x^{e_x}\). The filtration by powers of \(\pi_x\) and (X.2) give

Here \(e_x\) is positive. In particular every fibre is nonempty, since \(B\) is free of positive rank; its nonzero finite-dimensional quotient has a maximal ideal. The generic fibre is nonempty too, so \(f\) is surjective. Formula (X.8) also shows

first for points and then for any divisor. Pullback of the local frames in X.2 identifies \(f^*\mathcal O(D)\) with \(\mathcal O(f^*D)\), so it is also the line-bundle degree formula once degree is shown well-defined. These are the local multiplicity arguments of The multiplicative group on a curve, Proposition 8.1, with residue-field factors included.

Apply this to \(f_a\). Its fibres at zero and infinity have the same weighted degree \(d\); their difference is \(\operatorname{div}(a)\), by (X.1). Constants have zero divisor. Thus every principal divisor has degree zero. Formula (X.5) now makes degree an intrinsic function of the line bundle, and (X.9) applies to every line bundle, including negative degree. No separability of the extension \(K(X)/K(Y)\) was used.

Finally these arguments give the affine complement used in Q.1. The point exact sequences, W.4 finiteness and X.2 yield (Q.2) by adding or removing finitely many points. For \(m\geq g+1\), that formula gives a nonconstant section of \(\mathcal O_C(mp)\). Viewed as a rational function, it has no pole except possibly \(p\), and it must have that pole because the global regular functions are \(k\). X.3–X.4 give a finite surjective map to \(\mathbf P^1_k\). The inverse image of \(\mathbf A^1_k\) is exactly \(C\setminus\{p\}\), hence is affine by finiteness. Fibres over the infinitely many points of \(\mathbf P^1(k)\) also show that \(C(k)\) is infinite. The degree argument precedes this affine-complement deduction, so the latter has not been assumed in proving (Q.2).

Appendix Y. From local growth to smooth curve parameters

The curve arguments need regular local rings to be domains, polynomial local rings to be regular, and a precise dimension count after independent equations are imposed. The common-zero argument in R.1 also needs a height bound. We prove these statements together. All rings in a local dimension assertion are nonzero and Noetherian. Polynomial rings are Noetherian by the leading-coefficient argument in W.4; finite-module Artin–Rees and local separation are proved in L.1; we use the determinant form of Nakayama proved in U.1.

Y.1. Counting the layers of a finite module

Let \((R,\mathfrak m,\kappa)\) be local and let \(I\) have radical \(\mathfrak m\). Since \(\mathfrak m\) has finitely many generators, powers of those generators lie in \(I\), and a sufficiently large power of \(\mathfrak m\) lies in \(I\). A finite module killed by a power of \(\mathfrak m\) has a finite filtration whose successive quotients are finite-dimensional \(\kappa\)-spaces. It therefore has finite length. We use length with its usual additivity: a composition series in a submodule, followed by lifts of a composition series in the quotient, is a composition series in the middle module. Independence of its length is proved by the maximal-submodule induction in Noetherian and Artinian rings, Theorems 3.2–3.3.

For a finite module \(M\), put

If \(I\) is generated by \(r\) elements, its graded ring is a quotient of \((R/I)[T_1,\ldots,T_r]\), with each variable of degree one. The graded module \(\bigoplus I^jM/I^{j+1}M\) is finite over that polynomial ring: generators of \(M/IM\), multiplied by monomials in the generators of \(I\), generate every layer.

Here is the graded calculation needed for (Y.1). For any finite graded module \(G\) over \(A[T_1,\ldots,T_r]\), where \(A\) has finite length over \(R\), set \(F_G(t)=\sum_j\ell_R(G_j)t^j\). The grading is bounded below and every component has finite length. Write \(L=\operatorname{ann}_G(T_r)\) and \(C=G/T_rG\). The degreewise kernel and cokernel sequence of multiplication by \(T_r\) gives

Both \(L\) and \(C\) are finite graded modules over the polynomial ring in \(r-1\) variables: finiteness of \(L\) follows from Noetherianness, and \(T_r\) kills both. For \(r=0\) only finitely many homogeneous degrees occur. Induction in (Y.2) proves that \(F_G(t)\) is a Laurent polynomial divided by \((1-t)^r\). For \(r>0\), the coefficient of \(t^n\) in \((1-t)^{-r}\) is \({n+r-1\choose r-1}\): it counts the ways to distribute \(n\) among \(r\) nonnegative exponents, or follows by multiplying geometric series. Summing coefficients adds one factor \((1-t)^{-1}\). Consequently (Y.1) agrees, for all sufficiently large \(n\), with a polynomial of degree at most \(r\). If \(r=0\), then \(I=0\), and the same assertion means a constant function.

For \(M\ne0\), Nakayama gives \(M/IM\ne0\). Thus the eventual polynomial is nonzero with positive leading coefficient. Denote its degree by \(d(M)\), and put \(d(0)=-\infty\). This degree is independent of \(I\): if \(I,J\) both have radical \(\mathfrak m\), choose positive \(a,b\) with \(I^a\subset J\) and \(J^b\subset I\). Then

Positive leading coefficients force equal degrees. Finally \(d(M)=0\) precisely when \(M\) has finite length. An eventually constant (Y.1) gives \(I^nM=I^{n+1}M\), hence \(I^nM=0\) by Nakayama. Conversely a composition series of length \(l\) over the local ring has all factors \(\kappa\), so \(\mathfrak m^lM=0\), making (Y.1) eventually constant.

Y.2. Exact sequences and a strict decrease

In a short exact sequence \(0\to N\to M\to Q\to0\), identify \(N\) with its image. L.1 gives, for a fixed \(c\) and \(n\ge c\),

The kernel of \(M/I^nM\to Q/I^nQ\) is \(N/(N\cap I^nM)\). Taking lengths therefore gives

The two bounding eventual polynomials have the same degree and the same leading coefficient. It follows that \(d(M)=\max(d(N),d(Q))\); in that largest degree their leading coefficients add. This also covers zero modules and degree zero. We only assert additivity of the largest-degree coefficient, not of the entire polynomial.

Suppose now that multiplication by \(x\in\mathfrak m\) is injective on nonzero \(M\). The quotient \(M/xM\) is nonzero by Nakayama. Apply (Y.4) to \(0\to M\xrightarrow{x}M\to M/xM\to0\). If the quotient had degree \(d(M)\), its positive leading coefficient would have to add to that of \(M\) without changing it, which is impossible. Hence

Y.3. Prime chains, parameters and height

Write \(\delta(R)\) for the least number of generators of an ideal with radical \(\mathfrak m\). We prove

First \(\dim R\le d(R)\), by induction on \(d(R)\). If this degree is zero, Y.1 gives \(\mathfrak m^n=0\) for some \(n\), so \(\mathfrak m\) is the only prime. For a chain \(\mathfrak p_0\subsetneq\cdots\subsetneq\mathfrak p_l\), there is nothing to prove if \(l=0\). Otherwise choose \(x\in\mathfrak p_1\setminus\mathfrak p_0\). In the local domain \(B=R/\mathfrak p_0\), multiplication by \(x\) is injective, and the positive-length chain ensures that \(x\) is a nonunit. Y.2 gives \(d(B/xB)<d(B)\le d(R)\). Lengths and growth degrees for these quotient rings are the same as for their underlying \(R\)-modules, since their simple factor is still \(\kappa\). The remaining chain has length \(l-1\) in \(B/xB\). Induction gives \(l-1\le d(B/xB)\le d(R)-1\), as required. This also proves that the local dimension is finite.

For clarity, the minimal-prime facts used next need no dimension theorem. Ascending chains of ideals make descending chains of closed subsets of \(\operatorname{Spec}R\) stabilize. A closed subset without a finite decomposition into irreducible closed subsets would have a minimal such counterexample; splitting it into two proper closed subsets gives a contradiction. Irreducible closed subsets are exactly \(V(\mathfrak p)\) for primes, so there are finitely many minimal primes and every prime contains one. Here the radical of an ideal is the intersection of primes containing it: if an element is not in the radical, localize at its powers and take a maximal ideal in the resulting nonzero quotient. This also proves the stated correspondence with irreducible closed subsets.

Induct on \(D=\dim R\) to prove \(\delta(R)\le D\). If \(D=0\), the only prime is \(\mathfrak m\), so the zero ideal already has radical \(\mathfrak m\). If \(D>0\), no minimal prime is \(\mathfrak m\). Choose \(x\in\mathfrak m\) outside their union. The finite avoidance argument is explicit: for incomparable primes \(\mathfrak p_i\), choose \(a_i\in\mathfrak m\setminus\mathfrak p_i\) and, for each \(j\ne i\), choose \(b_{ij}\in\mathfrak p_j\setminus\mathfrak p_i\). The sum \(\sum_i a_i\prod_{j\ne i}b_{ij}\) is outside each \(\mathfrak p_i\). A single prime uses the empty product. Every prime chain in \(R/xR\) lifts to a chain in \(R\) whose bottom contains a strictly smaller minimal prime, because that minimal prime does not contain \(x\). Therefore \(\dim(R/xR)\le D-1\). Induction produces at most \(D-1\) elements generating an ideal of definition in this quotient. Their lifts together with \(x\) give \(\delta(R)\le D\). Finally Y.1 bounds \(d(R)\) by the number of generators of any ideal of definition. These three inequalities prove (Y.6).

If \(\mathfrak p\) is minimal over an ideal generated by \(r\) elements in a Noetherian ring, localize at \(\mathfrak p\). The extended ideal has radical the maximal ideal: a smaller prime containing it would contradict minimality. Its \(r\) generators and (Y.6) give

For \(x\in\mathfrak m\), lift a minimal system of parameters of \(R/xR\) and adjoin \(x\). Equation (Y.6) gives \(\dim(R/xR)\ge\dim R-1\). If \(x\) avoids all minimal primes, the chain-prepending argument gives the reverse inequality. A nonzerodivisor does avoid them: in \(R_{\mathfrak p}\) for a minimal prime \(\mathfrak p\), the maximal ideal is the nilradical, so any element of it is nilpotent. Multiplication by a nonzerodivisor remains injective after localization; a nilpotent element cannot act injectively on this nonzero ring. We have proved

This proof does not need the theory of associated primes.

Y.4. Regular parameters and their initial forms

Nakayama identifies \(e=\dim_\kappa\mathfrak m/\mathfrak m^2\) with the least number of generators of \(\mathfrak m\). Thus (Y.6) gives \(\dim R\le e\). Suppose equality holds, and set \(D=e=\dim R\); this is the definition of a regular local ring. Choose minimal generators \(x_1,\ldots,x_D\). They give a graded surjection

For \(D=0\), Nakayama gives \(\mathfrak m=0\), so \(R=\kappa\). For \(D>0\), a nonzero homogeneous relation \(F\) of degree \(a>0\) in (Y.9) would bound \(H_{\mathfrak m,R}(n)\) by the number of monomials of degree less than \(n\) in the quotient by \(F\). Multiplication by \(F\) is injective in the polynomial ring over the field. For \(n\ge a\), that number is

which has degree at most \(D-1\). This contradicts (Y.6). Hence (Y.9) is an isomorphism.

L.1 separation shows that every nonzero element of \(R\) has a finite \(\mathfrak m\)-adic order. Its initial form is nonzero in the polynomial domain (Y.9). Products of nonzero initial forms remain nonzero, so orders add and products of nonzero elements do not vanish. Thus \(R\) is a domain. In particular \(x_1\) is a nonzerodivisor, and the order calculation gives

Indeed, if \(x_1b\in\mathfrak m^n\) with \(b\ne0\), additivity of order forces \(b\in\mathfrak m^{n-1}\). To compute the graded ring of \(R/x_1R\), a kernel class in degree \(n\) is represented by \(u=x_1b+v\), with \(u\in\mathfrak m^n\) and \(v\in\mathfrak m^{n+1}\). Equation (Y.11) identifies its initial form as a multiple of \(T_1\), and every such multiple occurs. Consequently

Its length-growth degree is \(D-1\), hence so is its dimension by (Y.6); its embedding dimension is also \(D-1\). The quotient is therefore regular. Repeating proves that the entire list is a regular sequence and that quotienting by its first \(c\) terms gives a regular local domain of dimension \(D-c\). Any independent list of classes in \(\mathfrak m/\mathfrak m^2\) extends to a basis, so the same conclusion applies to any such chosen list of elements.

We will also use the converse lifting step. If \(x\in\mathfrak m\) is a nonzerodivisor and \(R/xR\) is regular of dimension \(e\), then (Y.8) gives \(\dim R=e+1\). Lifts of its \(e\) maximal-ideal generators, together with \(x\), generate \(\mathfrak m\). Thus the embedding dimension is at most \(e+1\), and the general reverse bound proves regularity of \(R\). Apply this step repeatedly when the quotient is by a regular sequence.

Y.5. Polynomial local rings over any field

Let \(A\) be a Noetherian ring whose prime localizations are regular. For a prime \(\mathfrak q\) of \(A[t]\), put \(\mathfrak p=\mathfrak q\cap A\), \(R=A_{\mathfrak p}\) and \(B=A[t]_{\mathfrak q}\). Choose a regular system of parameters \(a_1,\ldots,a_d\) of \(R\), whose regular-sequence property was just proved. The map \(R\to B\) is flat: polynomial extension is free, localization is exact, and tensoring these operations preserves exact sequences. Hence multiplication by each \(a_i\) stays injective after the preceding ones are killed in \(B\). The quotients are nonzero, since the \(a_i\) belong to the maximal ideal of \(B\).

The last quotient is a local ring of \(\kappa(\mathfrak p)[t]\), either its fraction field or its localization at one monic irreducible polynomial \(h\). Euclidean division proves the description of the prime ideals of this one-variable ring. These two local rings are regular of dimensions \(\varepsilon=0\) and \(\varepsilon=1\), respectively: in the latter case the maximal ideal is generated by \(h\), and the only prime chain is \((0)\subsetneq(h)\). Repeated use of (Y.8) gives \(\dim B=d+\varepsilon\); the lifting step in Y.4 gives regularity of \(B\). Induction starting with a field proves regularity of all polynomial local rings, without a homological global-dimension theorem or a flat-local dimension formula.

The same induction calculates their dimensions. In the fraction-field case the residue field is \(\kappa(\mathfrak p)(t)\), adding one transcendence degree; in the irreducible-polynomial case it is finite algebraic over \(\kappa(\mathfrak p)\), adding none. Thus for any field \(K\) and any prime \(\mathfrak q\subset K[t_1,\ldots,t_N]\),

We also need the residue-field statement at closed points. The normalization construction of K.1 works over every field: a sufficiently separated choice of integer weights makes a selected highest weighted monomial unique, and division by its nonzero coefficient makes the resulting relation monic in the last variable. Neither step uses characteristic zero or an infinite field. Successively eliminating dependent generators makes any finite-type domain finite integral over a polynomial subring on algebraically independent elements. If the domain is itself a field, that polynomial subring must be a field: an integral equation for the inverse of a nonzero base element, multiplied by the appropriate power of that element, puts its inverse back in the base. A polynomial ring in a positive number of variables is not a field. Thus a field of finite type over \(K\) is finite algebraic over \(K\). Applied to a maximal quotient, this proves the assertion for closed points, including over imperfect fields. Over an algebraically closed \(K\), the residue field is \(K\). If an element \(f\) in a finite-type algebra is not nilpotent, its localization at \(f\) is nonzero and still of finite type; a maximal quotient there gives a closed \(K\)-point where \(f\ne0\). This is the strong Nullstellensatz used in R.1.

Y.6. Smooth presentations and the dimension used above

We finish by verifying the direction of the smoothness criterion actually needed in V and X. Suppose \(S\) is a finitely presented, formally smooth algebra over a field \(K\), and write \(S=P/I\), where \(P=K[t_1,\ldots,t_N]\). Formal smoothness applied to \(P/I^2\to S\) lifts the identity to a \(K\)-algebra section \(\sigma:S\to P/I^2\). The function

is a \(K\)-derivation, with values in the \(S\)-module \(I/I^2\), and sends \(f\in I\) to its class. Indeed multiplication of two differences vanishes modulo \(I^2\), giving the Leibniz rule. A derivation on a polynomial ring is determined by its values on the variables: induction on monomials gives \(\delta(p)=\sum_i (\partial p/\partial t_i)\delta(t_i)\). Therefore the map

has a left inverse supplied by \(\delta\). In particular it is a split injection. Its cokernel represents derivations of \(S\), because precisely the derivatives of the relations must vanish; this is the differential module \(\Omega_{S/K}\). Both \(I/I^2\) and that cokernel are finite projective, as direct summands of \(S^N\). Finite projective modules are locally free by the finite-flat local argument in U.1.

At a prime of \(S\) choose \(f_1,\ldots,f_c\in I\) whose classes are a basis of \((I/I^2)\) at that local ring. The split injection (Y.15) makes their gradient rows independent over the residue field; choose a nonzero \(c\)-column minor. At the polynomial prime \(\mathfrak Q\) above this point, Nakayama applied to the finite module \(I_{\mathfrak Q}/(f_1,\ldots,f_c)\) gives \(I_{\mathfrak Q}=(f_1,\ldots,f_c)\): its equality with \(I_{\mathfrak Q}\) times itself follows from the basis assertion modulo \(I^2\), and \(I_{\mathfrak Q}\) is inside the maximal ideal. Finite generation lets one invert a single element outside \(\mathfrak Q\) so that these generators and the nonzero minor work on a neighborhood. This supplies the required standard smooth presentation locally.

Now Y.5 makes \(P_{\mathfrak Q}\) regular. The classes of the \(f_i\) in its maximal ideal modulo its square are independent: their images under the differential map are the independent gradient rows, and the differential of a product of two maximal-ideal elements vanishes over the residue field. Y.4 therefore makes their quotient a regular local domain and gives

On this neighborhood the invertible minor identifies the differential module with a free module of rank \(N-c\), by solving the \(c\) independent linear relations. At a closed point, Y.5 makes the transcendence degree zero. Hence its local dimension is exactly that differential rank. Formal smoothness is preserved by field extension: restrict a lifting problem for \(S\otimes_K K'\) to \(S\), lift it there, and combine the lift with the given structural map from \(K'\). Finite presentation is also preserved. The entire argument consequently applies after any field extension, with no assumption on residue-field separability.

In V.3 the differential rank of the coefficient chart is \(d\), so every closed local ring on that chart is regular of dimension \(d\). This also gives dimension \(d\) of the chart: every prime chain extends to a maximal ideal, and a closed local ring has dimension \(d\). In X.1 the smooth curve has differential rank one after every field extension, so its closed local rings are one-dimensional regular domains. These are exactly the local inputs for the valuation and divisor calculations. Finally the strong Nullstellensatz in Y.5 and the height bound (Y.7) give the common-nonzero-zero assertion in R.1. Each application uses a proof given here, in addition to the previously established finite-module and normalization arguments.

Appendix Z. Splitting algebras and killing scalar two-cocycles

We prove the algebra and descent statements needed for (R.4). The fields to which R applies contain the algebraically closed field \(k\), and are therefore infinite. This lets us construct a separable splitting field by choosing one element with distinct eigenvalues. The final step turns a scalar two-cocycle into semilinear matrices and exhibits its coboundary after splitting the associated algebra.

Z.1. The finite simple-algebra calculation

Let \(A\) be a finite-dimensional simple algebra over a field \(F\). A nonzero left ideal of least positive dimension is a simple left module. The sum of all simple left ideals is a nonzero two-sided ideal: right multiplication maps any simple left ideal either to zero or isomorphically onto another simple left ideal. Simplicity makes that sum all of \(A\). Choose a direct sum of simple left ideals of maximal dimension. Every other simple left ideal either meets it trivially, so could be added, or is contained in it. Maximality thus makes the direct sum equal to \(A\).

All its simple summands have one isomorphism type \(S\). To see this, group a chosen decomposition by type. A homomorphism between nonisomorphic simple modules is zero, since any nonzero homomorphism is injective and surjective. Projections therefore show that any simple submodule of a given type lies in that type's group of summands. Right multiplication preserves each such group. Each is consequently a two-sided ideal, and simplicity permits only one. Write \({}_AA\simeq S^r\), and set \(E=\operatorname{End}_A(S)\). The same kernel-and-image argument makes \(E\) a division algebra. Matrices of endomorphisms give

The first equality sends an endomorphism to its value at \(1\): the endomorphism is right multiplication by that value, which reverses multiplication. Transposition identifies the opposite matrix algebra with matrices over the opposite division algebra. The center of \(M_r(D)\) is the scalar matrices over \(Z(D)\), as is checked by commuting with the matrix units and then the diagonal entries. Thus, if \(A\) is central over \(F\), so is \(D\). This proves the part of the finite Wedderburn theorem used below.

Central simplicity survives every extension \(K/F\). Here is the tensor argument, including infinite extensions. In a nonzero two-sided ideal of \(A\otimes_F K\), choose a nonzero element \(x=\sum_{i=1}^t a_i\otimes b_i\) of least tensor length, with the \(b_i\) linearly independent over \(F\) and \(a_1\ne0\). Since the ideal \(Aa_1A\) is all of \(A\), choose \(u_j,v_j\) with \(\sum_j u_ja_1v_j=1\). Then \(y=\sum_j(u_j\otimes1)x(v_j\otimes1)\) has first coefficient \(1\), so is nonzero. Its commutator with \(a\otimes1\) has at most \(t-1\) terms and must vanish by minimality. Independence of the \(b_i\) makes all remaining coefficients central in \(A\), hence scalar. Thus \(y=1\otimes b\) for nonzero \(b\in K\). It is a unit, proving simplicity. For any central element, the same coefficientwise commutator argument puts it in \(1\otimes K\), proving centrality.

Over an algebraic closure \(\overline F\), a finite-dimensional division algebra has only scalar elements. An element satisfies a polynomial relation; factor that polynomial into linear factors over \(\overline F\). Evaluation at this one element respects their product, and a division algebra has no zero divisors, so one factor vanishes. Applying (Z.1) after scalar extension gives

We also need only the matrix case of inner automorphisms. For an automorphism of \(M_n(K)\), let \(f_{ij}\) be the images of its matrix units. Choose nonzero \(v\) with \(f_{11}v=v\), and put \(v_i=f_{i1}v\). The identities \(f_{ij}v_l=\delta_{jl}v_i\) show that the \(n\) vectors \(v_i\) are linearly independent, by applying \(f_{1j}\) to a proposed relation. They are a basis, and in that basis the \(f_{ij}\) are the ordinary matrix units. Thus the automorphism is conjugation by a change-of-basis matrix. A matrix commuting with all matrix units is scalar.

Z.2. A separable splitting field and the norm form

Assume \(F\) infinite and let \(D\) be central division, with \(\dim_F D=n^2\). Fix (Z.2) and an \(F\)-basis \(a_1,\ldots,a_{n^2}\). Its images \(B_i\) form an \(\overline F\)-basis of \(M_n(\overline F)\). For scalar coordinates \(z\), set \(B(z)=\sum_i z_iB_i\) and consider the polynomial

If the eigenvalues of \(B(z)\), counted with multiplicity, are \(\lambda_1,\ldots,\lambda_n\), this determinant is \(\prod_{i<j}(\lambda_j-\lambda_i)^2\). Indeed a matrix over an algebraically closed field can be made upper triangular by choosing an eigenvector and continuing on the quotient. Its power traces are \(\sum_l\lambda_l^{i+j}\); the displayed matrix is the product of the Vandermonde matrix with its transpose. The Vandermonde determinant is the indicated product: it vanishes whenever two variables coincide, has the same total degree as that product, and comparison of the leading monomial gives coefficient one. In particular (Z.3) is not the zero polynomial, since the \(B_i\) span a diagonal matrix with distinct entries. This works in every characteristic.

A nonzero polynomial over \(\overline F\) cannot vanish on every tuple from infinite \(F\). Induct on the number of variables, using the one-variable root bound and then a nonzero coefficient polynomial. Choose \(z\in F^{n^2}\) with \(\Delta(z)\ne0\), and let \(a=\sum_i z_i a_i\in D\). Its matrix has \(n\) distinct eigenvalues, so its minimal polynomial over \(\overline F\) is the product of the \(n\) distinct linear factors. The least degree of a relation among \(1,a,a^2,\ldots\) does not change under a field extension: ranks of matrices with entries in \(F\) do not change, as their nonzero minors and zero minors do not change. A linear system with coefficients in \(F\) that is soluble over \(\overline F\) is soluble over \(F\), by row reduction. Thus the unique monic minimal polynomial of \(a\) over \(F\) has degree \(n\) and those same distinct roots.

The commutative finite-dimensional algebra \(L=F[a]\) is a domain inside \(D\), hence a field: multiplication by a nonzero element is injective and therefore surjective as a linear map. We have proved that \(L/F\) is separable of degree \(n\). Regard \(D\) as a right \(L\)-space, of dimension \(n\). The map

is multiplicative because \(L\) is commutative. Its domain is simple by Z.1, and it preserves the identity, so is injective. Both sides have \(L\)-dimension \(n^2\), making it an isomorphism. A finite Galois extension \(E/F\) containing \(L\) therefore splits \(D\). Such an extension is obtained by adjoining all roots of its separable minimal polynomial. The elementary Galois facts used here and below are detailed in Z.3.

Choose a splitting \(\phi:D\otimes_F E\to M_n(E)\). For the same basis of \(D\), write \(C_i=\phi(a_i)\). The polynomial

is homogeneous of degree \(n\), and all its coefficients belong to \(F\). For \(g\in\operatorname{Gal}(E/F)\), transport \(1\otimes g\) across \(\phi\). Composing that semilinear automorphism with inverse entrywise \(g\) gives an \(E\)-algebra automorphism of matrices. Z.1 makes it inner, so \(g(C_i)\) are simultaneously conjugate to \(C_i\). Taking determinants gives \(g(N)=N\) coefficientwise. The fixed-field assertion in Z.3 puts those coefficients in \(F\).

The value of (Z.5) at an element of \(D\) is zero precisely when that element is not invertible. One direction follows by extending an inverse. Conversely, if the matrix is invertible, left multiplication by the element on \(D\) becomes invertible after extension to \(E\). Its determinant over \(F\) is consequently nonzero, so there is already a right inverse in \(D\). Injectivity of that left multiplication turns a right inverse into a two-sided inverse. This argument also works for a general finite algebra once a matrix splitting is given.

For an infinite \(C_1\)-field \(F\), (Z.5) rules out \(n>1\): it would be a degree-\(n\) homogeneous form in \(n^2>n\) variables, with a nonzero zero, contrary to division. Hence every central division algebra over \(F\) is \(F\), and (Z.1) makes every finite central simple algebra a matrix algebra. In particular its Brauer group is zero. R.1 proves the required \(C_1\) property for every field algebraic over the function field under consideration, so this conclusion holds for all those fields.

Z.3. Finite field automorphisms and semilinear descent

We record the field facts used in this calculation over infinite \(F\). An embedding of a field into an algebraically closed field extends over a finite algebraic extension by adjoining generators in succession and choosing a root of each transported minimal polynomial. If the extension is separable, the number of embeddings is its degree: at each step the distinct roots count the degree, and the counts multiply. For two separable generators \(a,b\), only finitely many \(c\in F\) make two distinct embeddings take the same value on \(a+cb\). Choose \(c\) outside that set. The element \(a+cb\) then has as many distinct conjugates as the degree of \(F(a,b)/F\), so it generates this extension. Induction gives the primitive-element assertion for every finite separable extension over infinite \(F\).

For a finite splitting field \(E/F\) of separable polynomials, every embedding permutes their roots and thus maps \(E\) onto itself. There are \([E:F]\) such automorphisms. Their fixed field is \(F\): if \(x\in E\setminus F\), its separable minimal polynomial has another root, and the corresponding embedding of \(F(x)\) extends to an automorphism of \(E\) moving \(x\). More generally, if \(H\) is any subgroup of this finite automorphism group and \(K=E^H\), a primitive generator \(a\) for the separable extension \(E/K\) satisfies \(\prod_{h\in H}(T-h(a))\in K[T]\). Thus \([E:K]\le |H|\), while its \(|H|\) distinct automorphisms give the reverse inequality. Hence \(\operatorname{Gal}(E/E^H)=H\) and \([E:E^H]=|H|\). Normal subgroups give normal fixed-field extensions by the same conjugation action. These arguments give the finite fixed-field statements needed for Galois splitting and finite quotient cochains.

Distinct automorphisms \(g:E\to E\) are linearly independent as functions with coefficients in \(E\). If a relation of least positive length existed, all its coefficients would be nonzero. Choose an element \(b\) on which its first automorphism and another differ. Subtract the first automorphism's value at \(b\) times the relation at \(x\) from the relation at \(bx\). One term disappears and another remains, giving a shorter nonzero relation. A one-term relation is impossible. This proves independence.

Let \(Q=\operatorname{Gal}(E/F)\), and let \(V\) be a finite-dimensional \(E\)-space with a genuine semilinear \(Q\)-action \(T_g\). Its fixed vectors span \(V\) over \(E\). Otherwise choose a nonzero \(E\)-linear functional \(\lambda\) annihilating their span. For every \(v\in V\) and \(b\in E\), the vector \(\sum_g g(b)T_g(v)\) is fixed, by reindexing after applying any \(T_h\). Hence

Independence of automorphisms forces each coefficient to vanish; the coefficient for the identity gives \(\lambda(v)=0\) for every \(v\), a contradiction. Choose an \(E\)-basis consisting of fixed vectors. A vector in that basis is fixed exactly when each coefficient lies in \(E^Q=F\). Consequently multiplication gives an isomorphism

When \(V\) is an algebra and the action is by semilinear algebra automorphisms, this map is also multiplicative and unital, so it is algebra descent. This finite-dimensional proof needs neither a trace-dual basis nor a normal-basis theorem.

The same independence proves scalar Hilbert 90 in the convention used for cochains. For \(b_{gh}=b_g g(b_h)\), choose \(x\in E\) such that \(u=\sum_g b_g^{-1}g(x)\ne0\), possible by independence. Applying \(h\), the cocycle identity and a change of index give \(h(u)=b_hu\). Thus \(b_h=h(u)/u\). Continuous cocycles on an absolute Galois group reduce to this finite calculation by the finite-quotient comparison in R.3.

Z.4. A scalar two-cocycle becomes a coboundary

Let \(F\) be one of the fields in R.1, and fix a continuous scalar two-cocycle on \(G_F\). By the continuous-cochain finite-quotient comparison (AA.4), it is represented over some finite Galois \(E/F\) by \(c:Q\times Q\to E^\times\), where

We can arrange \(c(1,g)=c(g,1)=1\). In fact (Z.8) gives \(c(1,g)=a\) independent of \(g\), and \(c(g,1)=g(a)\). Multiplying by the coboundary of the constant one-cochain \(a^{-1}\) gives those normalizations.

On the \(E\)-space \(W\) with basis \((e_h)_{h\in Q}\), define invertible semilinear maps

Substitution in (Z.8) gives \(T_gT_h=c(g,h)T_{gh}\), and \(T_1=1\). Thus conjugation by \(T_g\) is a genuine semilinear group action on \(\operatorname{End}_E(W)\), since its scalar multiplier cancels. Let

Z.3 gives \(E\otimes_F A=\operatorname{End}_E(W)\). The algebra \(A\) is central simple over \(F\). A nonzero proper two-sided ideal would stay nonzero and proper after extending scalars, by vector-space dimensions, contradicting simplicity of a matrix algebra. For completeness, that simplicity follows by multiplying any nonzero matrix entry in an ideal by matrix units and its scalar inverse to obtain every matrix unit. A central element of \(A\) commutes after extension with every endomorphism of \(W\), since \(A\) spans them over \(E\); it is scalar. Invariance then puts that scalar in \(F\).

By Z.2 and R.1, \(A\simeq M_m(F)\), with \(m=|Q|\) by dimensions. Choose its matrix units inside \(\operatorname{End}_E(W)\). The basis construction in Z.1 puts these units into their usual form. In this basis each \(T_g\) fixes them under conjugation. Write \(T_g=B_g g\), with \(g\) acting entrywise on coordinates. The matrix units have entries in \(F\), so \(B_g\) commutes with all of them and is scalar, say \(B_g=b_g I\). Comparing multipliers gives

Thus \(c\) is a coboundary at this same finite stage. Inflating proves \(H^2(G_F,(F^s)^\times)=0\); the identical argument applies to every field algebraic over \(F\). Along with Z.2, this gives both vanishings in (R.4). The full Brauer–cohomology isomorphism remains the theorem in R.2's programme reference; the vanishing used in R.3 follows already from the explicit scalar identity (Z.11).

Appendix AA. Continuous cochains and the Sylow reduction

We supply the group-theoretic steps used in R.3–R.4. In particular the Sylow subgroup can have infinite index in the profinite group. The transfer calculation below is performed in a finite quotient containing both a cocycle and a primitive for its restriction.

AA.1. Compact finite limits and fixed fields

A product of finite discrete spaces is compact. Here is the form of the proof we need. Extend a family of closed sets with the finite intersection property to an ultrafilter, using Zorn's lemma. Exactly one cell of each finite coordinate partition lies in that ultrafilter: two disjoint cells cannot both belong, and if none belonged their complements would have empty intersection. These selected coordinates define a point. Every basic neighborhood of that point lies in the ultrafilter, so the point lies in every original closed set, by closedness. Compatibility equations in a product of finite sets are closed conditions. If every finite collection of such equations can be satisfied, their whole system therefore has a solution. This also proves that a directed inverse system of nonempty finite sets with surjective transition maps has a nonempty limit, and that each limit projection is surjective, by imposing one prescribed coordinate value.

We use a profinite group in its inverse-limit topology. It is compact and Hausdorff, and the open normal subgroups form an identity-neighborhood basis. Each has finite index. A continuous map from a finite product of profinite groups to a discrete set has finite image and factors through one finite quotient in each coordinate: choose finitely many basic open rectangles on which it is constant and intersect their open normal coordinate subgroups. If finitely many values belong to a discrete continuous module, the subgroup can be shrunk further to fix them all. For a closed subgroup \(H\subset G\), its topology is induced from \(G\), so the subgroups \(H\cap U\), for open normal \(U\subset G\), give the corresponding basis in \(H\).

For clarity, the Galois group assertions needed in R.3 are Profinite groups and infinite Galois theory, Section 3, with the finite algebra proved in Section 0 and also in Z.3. Every finite subset of a separable closure lies in a finite Galois extension. Compatible automorphisms of those finite fields define an automorphism of their union, giving the inverse-limit description. Restrictions to finite Galois fields are surjective: the successive-root extension of an embedding extends through an algebraic closure by a maximal-extension argument, and normality makes it preserve the separable closure. The image of that extended embedding is algebraically closed and contains the base field, so contains all elements algebraic over that base; thus it is an automorphism. This is the extension argument written out in the cited section.

Here is the closed-subgroup consequence in the exact form we use. For closed \(H\subset G_F\), put \(L=(F^s)^H\). If \(g\notin H\), compactness and the open normal basis give a finite quotient in which its image lies outside the image of \(H\). Finite Galois correspondence supplies an element of the latter image's fixed field moved by \(g\). That element belongs to \(L\), so the subgroup fixing \(L\) is exactly \(H\). The field \(F^s\) is algebraic separable over \(L\): the minimal polynomial over \(L\) of any of its elements divides its separable polynomial over \(F\). It is already separably closed, hence is a separable closure of \(L\). Its Krull topology over \(L\) is the induced topology, since both have the finite-element stabilizers as a basis. Consequently \(G_L=H\) as a topological group.

AA.2. Why these continuous cochains compute cohomology

For a discrete continuous \(G\)-module \(M\), use equivariant homogeneous cochains

with the alternating omission differential. The two orders of omitting a pair of entries have opposite signs, so \(d^2=0\). An equivariant cochain is determined by the arbitrary continuous function \(f(1,x_1,\ldots,x_n)\); its recovery formula is \(f(g_0,\ldots,g_n)=g_0 f(1,g_0^{-1}g_1,\ldots,g_0^{-1}g_n)\). Thus a surjection of discrete modules induces a surjection of cochain groups: lift the finitely many values of that arbitrary function, using its clopen fibres. A short exact sequence of modules gives a short exact sequence of cochain complexes. The cycle-lifting argument in T.4 gives its natural long exact sequence. Degree-zero cohomology is \(M^G\).

For an abelian group \(A\), put

This is discrete, by finite-quotient factorization. The functor is exact, since a function's finitely many values can be lifted through any surjection. Evaluation at \(1\) identifies \(G\)-maps \(M\to I_G(A)\) with abelian-group maps \(M\to A\); the inverse sends \(\psi\) to \(m\mapsto[t\mapsto\psi(tm)]\). These are inverse by equivariance, and the displayed functions are continuous because the orbit of \(m\) is finite. Thus \(I_G\) is right adjoint to the exact forgetful functor and preserves injectives. The map \(m\mapsto[t\mapsto tm]\) embeds \(M\) equivariantly in \(I_G(M)\).

There are enough injective abelian groups. One direct construction embeds any \(A\) in a divisible group. At one stage adjoin a symbol \(e_{a,n}\) for every \(a\in A\), \(n>0\), with relation \(n e_{a,n}=a\). The map from \(A\) into this quotient is injective: a finite sum of these relations with zero symbol coordinates has every coefficient zero. Repeat for all elements of the new group and take the union over countably many stages. Every element then has a root of every positive order, so the union is divisible. A divisible group \(D\) is injective: extend a map from a subgroup maximally by Zorn. To adjoin an omitted element \(x\), if no positive multiple of \(x\) lies in the current subgroup, choose its image arbitrarily. Otherwise choose the least positive \(n\) with \(nx\) in that subgroup and solve \(ny=f(nx)\) in \(D\). The formula \(f(b+mx)=f(b)+my\) is well-defined, since every relation has \(m\) a multiple of \(n\). Either choice contradicts maximality unless the domain is the whole group. Embedding the underlying group of \(M\) in such a \(D\) and using \(M\to I_G(D)\) proves enough injectives for discrete \(G\)-modules.

Positive-degree cochain cohomology of \(I_G(A)\) is zero. Evaluation at \(1\) takes an equivariant cochain with these coefficients to an arbitrary continuous function \(h:G^{n+1}\to A\). Its inverse is \(f(g_0,\ldots,g_n)(t)=h(tg_0,\ldots,tg_n)\). These formulas are continuous also as maps into the discrete function group: a finite quotient through which \(h\) factors works uniformly for the argument tuples. The differential becomes ordinary omission on \(h\). Inserting \(1\) as the first argument contracts the augmented omission complex; expansion gives \(ds+sd=1\). Therefore

For an injective \(G\)-module \(J\), the embedding \(J\to I_G(J)\) splits equivariantly. Its positive cochain cohomology is a direct summand of zero, so is zero. This proves that the cochain groups compute the right derived functors of invariants. More explicitly, embed \(M\) in an injective \(J\) with quotient \(N\). Both the cochain and injective-resolution long exact sequences identify degree one with \(\operatorname{coker}(J^G\to N^G)\), and identify degree \(q>1\) with degree \(q-1\) for \(N\). Iterating gives the comparison in every degree; compatible choices for maps are constructed as in T.4, and T.3's resolution homotopies give independence of choices. We can henceforth write these cohomology groups as \(H^q(G,M)\).

The finite-factorization observation in AA.1, applied to the arbitrary inhomogeneous coordinates of a cochain, gives

Indeed every cochain and every equality of cochains occurs at a common finite quotient after shrinking \(U\) to fix their values. Transition maps are inflation with inclusion of coefficient invariants. Kernels and images commute with this filtered union: a cycle equation or a boundary equation involves finitely many elements and holds at one stage. The same finite-value argument shows that cochain cohomology commutes with filtered colimits of coefficient modules. Lift finitely many values to a common stage; any equality between two such lifts holds at a later common stage. These arguments prove both assertions at the level of complexes before passing to cohomology.

AA.3. Constructing compatible Sylow subgroups

Let a finite group \(Q\) have order \(\ell^a m\), with \(\ell\nmid m\). The number of subsets of \(Q\) with \(\ell^a\) elements is not divisible by \(\ell\). In fact the coefficient of \(T^{\ell^a}\) in

is \(m\) modulo \(\ell\). The congruence follows by iterating \((u+v)^\ell=u^\ell+v^\ell\) over \(\mathbf F_\ell\). Left translation acts on these subsets. Some orbit has cardinality prime to \(\ell\), since otherwise their total cardinality would be divisible by \(\ell\). Its stabilizer \(S\) has order divisible by \(\ell^a\), by the orbit-stabilizer coset count. On the selected subset, left translation by \(S\) is free, so \(|S|\) divides its cardinality \(\ell^a\). Hence \(|S|=\ell^a\), proving existence of a Sylow subgroup.

Any \(\ell\)-subgroup \(P\) of \(Q\) acts on the \(m\) cosets \(Q/S\). Its orbits have \(\ell\)-power size. Because \(m\) is prime to \(\ell\), a fixed coset exists, and it gives \(P\subset qSq^{-1}\). Thus every \(\ell\)-subgroup is contained in a Sylow subgroup and all Sylow subgroups are conjugate. If \(N\) is normal in \(Q\), then \(S\cap N\) is Sylow in \(N\): a Sylow subgroup of \(N\) is contained in a conjugate of \(S\), and normality brings that conjugate subgroup back into \(S\cap N\). Counting orders now shows that the image of \(S\) is Sylow in \(Q/N\). Every Sylow subgroup of \(Q/N\) is obtained this way, by conjugacy and lifting its conjugating element.

For profinite \(G\), the finite sets of Sylow subgroups of \(G/U\) therefore form an inverse system under quotient images. Any finite collection of compatibility conditions is satisfied by taking a Sylow subgroup in a common finer quotient. AA.1 compactness supplies compatible choices \(P_U\). Put

The chosen quotient maps on \(P_U\) are surjective. The prescribed-coordinate form of compactness shows that \(P\) maps onto every \(P_U\). It is closed and is their inverse limit, so is pro-\(\ell\). This is the Sylow pro-\(\ell\) subgroup used in R.3.

AA.4. Restriction detected by a finite transfer

First let \(Q\) be finite and \(S\subset Q\). Choose one representative in each orbit of left multiplication by \(S\) on \(Q\), including \(1\) for the orbit \(S\). Define \(r:Q\to S\) by \(r(st)=s\) in this choice. It is \(S\)-equivariant and restricts to the identity on \(S\).

Suppose a homogeneous \(Q\)-cocycle \(c\) of degree \(n>0\) restricts to \(db\) on \(S\). We first construct a primitive on all tuples from \(Q\), equivariant only for \(S\). Regard \(u=r\) as a map \(Q\to Q\). For a degree-\(n\) cochain \(f\) on \(Q\), set

The omission expansion gives \(dH+Hd=1-u^*\). Terms deleting an entry before or after the place where the two lists meet cancel with the same deletion in the neighboring sum; the two end terms are \(f(x_0,\ldots,x_n)\) and \(-f(ux_0,\ldots,ux_n)\). This also verifies the sign in degree zero, where \(Hd f(x)=f(x)-f(ux)\). Both maps are \(S\)-equivariant. Therefore

Average this primitive over the cosets \(tS\in Q/S\):

Replacing \(t\) by \(ts\) leaves the summand unchanged by \(S\)-equivariance. Translating all arguments by an element of \(Q\) and reindexing the cosets proves \(Q\)-equivariance. This sum commutes with omission, and every summand applied to the \(Q\)-equivariant cocycle \(c\) is \(c\). Thus \(d\operatorname{Tr}B=[Q:S]c\). If \(M\) is \(\ell\)-primary and \([Q:S]\) is prime to \(\ell\), multiplication by that index is an automorphism of \(M\), its cochains and its cohomology. Its inverse on an element killed by \(\ell^r\) is multiplication by the inverse residue modulo \(\ell^r\), independent of a larger choice of exponent. Restriction to \(S\) is consequently injective on cohomology with these coefficients.

Now let \(P\) be (AA.6) inside profinite \(G\), and let a continuous cocycle \(c\) restrict to a boundary on \(P\). Formula (AA.4) places \(c\) at a finite quotient. The primitive on \(P\) also has finitely many values and factors through a finite quotient of \(P\). Shrink the open normal subgroup \(U\subset G\) so that \(P\cap U\) suffices for that factorization and \(U\) fixes both finite sets of values. This is possible using the induced topology described in AA.1. Both the cocycle and its restricted primitive now lie over \(G/U\), with coefficients in \(M^U\). The image \(P_U\) is Sylow there, so the finite calculation kills the class already at that stage. Degree zero is the inclusion of invariant groups. We have proved

No transfer of an infinite index was used.

AA.5. From one coefficient to all higher degrees

Suppose \(P\) is pro-\(\ell\) and \(H^2(P,\mathbf F_\ell)=0\), with trivial action. For any finite nonzero \(\ell\)-primary \(P\)-module \(M\), its subgroup \(M[\ell]\) is nonzero. Its action factors through a finite \(\ell\)-group. All nonsingleton orbits have cardinality divisible by \(\ell\), while \(|M[\ell]|\) is divisible by \(\ell\); the fixed set contains zero and therefore has at least \(\ell\) elements. A nonzero fixed element spans a trivial \(\mathbf F_\ell\)-line. Repeat on the smaller quotient. This gives a finite filtration by trivial lines, and the long exact cochain sequences give \(H^2(P,M)=0\).

Every discrete \(\ell\)-primary module is the union of its finite \(P\)-stable submodules. Each element has finite orbit, by its open stabilizer and compactness; the subgroup generated by finitely many such orbits is finite, since its finitely many generators have bounded \(\ell\)-power order. Formula (AA.4)'s coefficient-colimit assertion extends the degree-two vanishing to every such module.

Embed \(M\) in \(I_P(M)\). Both that module and its quotient are \(\ell\)-primary: each continuous function has finite image, hence is killed by one power of \(\ell\). Acyclicity (AA.3) and the long exact sequence give \(H^{r+1}(P,M)=H^r(P,I_P(M)/M)\) for \(r\ge1\). Repeat on the quotient to reduce any degree greater than two to the proved degree-two case. Thus all degrees greater than one vanish. Combining this result with (AA.10), and then taking finite sums and filtered unions of primary parts, proves the torsion assertion (R.9).

For the rational coefficient step in R.4, a finite group \(Q\) has the homogeneous cochain operator \(sf=\sum_{u\in Q}f(u,-)\). Reindexing proves equivariance, and the insertion expansion gives \(ds+sd=|Q|\) in positive degrees. Thus \(|Q|\) kills positive finite-group cohomology. By (AA.4), every positive profinite cohomology class is torsion. For a rational vector space of coefficients the cochain groups and their cohomology are rational vector spaces, so these torsion groups are zero. This proves exactly the rationalization input in (R.10)–(R.11).

Appendix AB. From field cohomology to the divisor sequence

The calculation in R passes from continuous Galois modules to sheaves on a field, and then from rational units to units on a curve. We give the exact field, slice and valuation arguments for those passages. Throughout the curve calculation, the ground field is the algebraically closed field \(k\) of R; the field calculation also applies to any field \(F\).

AB.1. The field objects of the étale site

An étale affine chart has a finitely presented formally étale coordinate algebra. Recall that formal étaleness means existence and uniqueness of a lift across a square-zero ideal. It implies formal smoothness and vanishing of the differential module: two lifts differ by a derivation, and the square-zero algebra \(B\oplus N\) tests all derivations into every \(B\)-module \(N\). Conversely a standard square presentation with invertible Jacobian has that unique lifting property, by correcting the tentative variables with the inverse Jacobian. These are the presentations obtained by the argument in Y.6 when its differential rank is zero.

Let \(B\) be such an algebra over \(F\), and put \(C=B\otimes_F\overline F\). Formal étaleness and finite presentation survive field extension: restrict a lifting problem to \(B\), solve it, and combine its lift with the given map from \(\overline F\); uniqueness is preserved as well. At each maximal ideal of \(C\), Y.6 gives a regular local ring of dimension zero, because the differential module is zero. That local ring is \(\overline F\): its maximal ideal has no generators and is zero by Nakayama, and its residue field is \(\overline F\) by Y.5.

Every prime of \(C\) is maximal. Otherwise a strict prime inclusion ending in a maximal ideal would remain strict in that localization, contradicting dimension zero. There are finitely many such primes, since a Noetherian ring has finitely many minimal primes. The ring is reduced: a nilpotent maps to zero in every maximal localization, hence is zero. Indeed, a nonzero element has a proper annihilator contained in a maximal ideal, and cannot vanish in that localization. The distinct maximal ideals are comaximal and their intersection is the nilradical, so the Chinese remainder map gives

The Chinese remainder assertion follows for two comaximal ideals from \(1=a+b\), with one term in each ideal, and then by induction. Extension of scalars preserves linear independence over a field, so (AB.1) makes \(B\) finite-dimensional over \(F\). It is reduced by its injection into \(C\). In a finite-dimensional algebra every prime quotient is a field: multiplication by a nonzero element in that domain is an injective linear map and therefore surjective. The same finite-prime and Chinese remainder argument writes \(B\) as a finite product of finite fields \(E_i/F\).

Each \(E_i/F\) is separable. If some element had an inseparable minimal polynomial, its subalgebra tensored with \(\overline F\) would contain a nonzero nilpotent. Tensoring its injection into \(E_i\) preserves injectivity, contradicting (AB.1). Conversely a finite separable extension is generated by one element, by the finite Galois algebra in Z.3 and Profinite groups and infinite Galois theory, Section 0. For its separable minimal polynomial \(p\), the derivative at the generator is invertible. The correction \(t\mapsto t-p(t)/p'(t)\) lifts a root uniquely across a square-zero ideal. Thus the extension is étale; finite products and arbitrary disjoint unions are étale locally on their components.

Every point of an étale scheme over \(F\) therefore has an open neighbourhood equal to the spectrum of a finite separable field. The scheme is a disjoint union of these open one-point schemes. It is quasi-compact precisely when the union is finite. This proves the field-fibre description in (R.12), including the empty fibre, after base change to the generic point.

AB.2. Descent on a finite Galois cover

Put \(G=G_F\). The preceding field description and AA.1 identify finite étale schemes \(V/F\) with finite continuous \(G\)-sets \(S_V=\operatorname{Hom}_F(\operatorname{Spec}F^s,V)\). For a connected object \(\operatorname{Spec}L\), choosing an embedding in \(F^s\) identifies this set with \(G/G_L\). An equivariant map of such orbits is a coset whose stabilizer contains the source stabilizer, and hence is exactly an embedding of the corresponding fixed fields in the opposite direction. Extension of embeddings and the closed fixed-field statement are proved in AA.1. This verifies full faithfulness, not just the correspondence on objects.

Products and covers become products and jointly surjective maps of these sets. To check this explicitly, place the finitely many field extensions in one finite Galois extension \(E/F\). A separable polynomial splits there with distinct roots; its Chinese remainder decomposition makes every finite étale algebra a product of copies of \(E\). Maps and products are then maps and products of finite indexing sets. A map between connected field objects is surjective, and is surjective on their embedding orbits by extension of embeddings. This also checks the assertion for covering families, component by component.

For a sheaf \(\mathscr F\) on this site define

Restrictions between field objects are injective: the field extension is a cover, and the sheaf separatedness axiom detects equality. Field automorphisms act compatibly on this colimit. Every representative at \(E\) is fixed by \(G_E\), so the resulting action on \(M\) is continuous.

For a finite separable \(L\subset F^s\), restriction identifies

Injectivity follows from the same covering argument. To prove surjectivity, represent a fixed element at a finite Galois \(E/F\) containing \(L\). Its section \(t\) is fixed by \(\operatorname{Gal}(E/L)\): equality in the colimit implies equality at \(E\), since its map into the colimit is injective. The two restrictions of \(t\) to \(E\otimes_L E\) therefore agree. In fact

is the Chinese remainder isomorphism for a primitive element's distinct roots. The sheaf axiom for \(\operatorname{Spec}E\to\operatorname{Spec}L\) descends \(t\) uniquely, proving (AB.3).

Conversely a continuous discrete \(G\)-set \(M\) gives a sheaf by \(V\mapsto\operatorname{Hom}_G(S_V,M)\): functions descend along a surjection precisely when they agree on its fibre product. Its colimit (AB.2) recovers \(M\), because each element has an open stabilizer containing some open normal \(G_E\). Formula (AB.3), and products over field components, show that the constructions are naturally inverse. This is the field-topos equivalence of Galois cohomology and the étale cohomology of a field, Theorem 1.1, expressed directly with finite Galois covers.

The constructions preserve abelian group operations, so give an equivalence of abelian categories. An equivalence preserves kernels, cokernels and injectives, by their universal properties and the extension definition of injectivity. Sections at \(F\) are invariants. Their injective-resolution complexes therefore agree. For the units sheaf, (AB.2) is exactly \((F^s)^\times\), since every element and its inverse occur in a finite Galois extension. Thus

The same reasoning gives the additive group, roots of unity and constant coefficients with their actual actions. For algebraically closed \(F\), the group is trivial and the sheaf category is the category of sets; its abelian section functor is exact.

AB.3. Exact inverse images and the slice calculation

We spell out the site operations used in R.5. Étale presentations are stable under base change and composition by their finite presentation and unique square-zero lifting property. Their diagonals are open immersions. In affine coordinates, the kernel \(I\) of \(B\otimes_A B\to B\) is finitely generated, and \(I/I^2=\Omega_{B/A}=0\). The latter equality sends \(b\otimes1-1\otimes b\) to \(db\); the product rule gives its universal derivation property. A finitely generated ideal with \(I=I^2\) is generated by an idempotent. Indeed, write a finite generator column \(x\) as \(Tx\) with entries of \(T\) in \(I\). The adjugate identity gives \(\det(1-T)I=0\). Write \(\det(1-T)=1-e\), where \(e\in I\). Then \((1-e)e=0\) and \(x=ex\), so \(e^2=e\) and \(I=(e)\). The quotient is the localization at \(1-e\), hence defines an open immersion. These affine diagonal charts cover the diagonal. A section of an étale map is a base change of its open diagonal, hence is open; factoring a morphism between étale schemes into its graph and a projection now shows that this morphism is étale too.

For any scheme map \(f:Y\to X\), write \(Y_V=Y\times_XV\). Base change preserves covering families, so \((f_*\mathscr F)(V)=\mathscr F(Y_V)\) is a sheaf. Its left adjoint is the sheafification of

The indexing uses restriction toward a refinement \(V'\to V\). It is filtered: \(X\) gives an object, a fibre product gives a common refinement, and the open equalizer of two maps gives a refinement equalizing them. The latter is open by the diagonal calculation just proved; the map from \(U\) factors through its base change by the required equality. Filtered colimits of sets and groups preserve finite limits, as is checked by placing finitely many representatives and their equalities at one common stage. Sheafification preserves finite limits by T.1. Formula (AB.6) is therefore left exact. The universal properties of its colimit and sheafification give the adjunction with \(f_*\); hence it also preserves cokernels and is exact on abelian sheaves. Consequently \(f_*\) preserves injectives.

For an object \(V\) of a site, restriction to its slice has the exact left adjoint which sheafifies

Adjunction follows by specifying the map on each summand. Injectivity is sectionwise injectivity in each summand. An epimorphism lifts a finite sum locally by choosing a common refinement for the finitely many local lifts; T.1 sheafification gives exactness. Slice restriction itself is exact, since local lifts are taken over that same slice object. It therefore carries an injective resolution to an injective resolution on the slice.

Apply this to an injective resolution \(\mathscr F\to I^\bullet\) on \(Y_{\rm ét}\). On the slice over \(Y_V\), its section complex is \(I^\bullet(Y_V)\). Exact sheafification commutes with kernels and images, so

Here \(a\) denotes associated sheaf. This is the direct resolution calculation in Pushforward, pullback and finite morphisms, Lemma 2.1; its proof uses neither higher-cohomology continuity nor a strict-local limit. For an étale object \(Y_V\), its small étale site is the slice: composition and the graph argument above identify the objects and maps, and the covers are the same.

AB.4. Uniformizers on étale curve neighbourhoods

Let \(U\to X\) be an étale chart of finite presentation over the smooth curve of R.5. It is smooth over \(k\): lift a map from \(X\) first, then use formal étaleness to lift the map from \(U\). Moreover

For completeness, a derivation from the base algebra into any module on the chart extends uniquely to the chart algebra. Apply the unique lifting property to its split square-zero algebra with structural map \(a\mapsto(a,da)\). This proves (AB.9) by the representing property of differentials. Its rank is one. By Y.6 the local rings are regular, and the closed local rings have dimension one. Distinct irreducible components cannot meet, since their meeting would give a local ring with two minimal primes instead of a domain. There are finitely many components on a quasi-compact chart. Each is an integral smooth curve; X.1 makes its closed local rings discrete valuation rings.

A closed point \(y\in U\) has residue field \(k\), and its image \(x\in X\) is closed: the induced quotient of an affine coordinate ring into \(k\) has image \(k\). Let \(\pi\) be a parameter at \(x\). Its differential is a basis of the cotangent space at \(x\). To see the identification, a \(k\)-derivation from its local ring to \(k\) is the same as a linear form on \(\mathfrak m_x/\mathfrak m_x^2\): subtract the residue scalar, use the linear form, and the product rule follows because products of two maximal-ideal elements vanish there. Formula (AB.9) carries this basis to the cotangent space at \(y\). Nakayama therefore gives

In particular each component maps nonconstantly to \(X\). If it mapped to a closed point, that point's parameter would pull back to zero, contradicting (AB.10) and dimension one. Its generic field is a finite separable extension of \(K=k(X)\), by AB.1 applied to the generic fibre. Every local map between these curve neighbourhoods has ramification index one: a parameter remains a parameter, a unit remains a unit, and \(u\pi^m\) keeps the same valuation. This proves compatibility of valuation with étale restriction.

It also proves the strict-local ring assertion mentioned in R.5. In the filtered system of pointed local étale neighbourhoods at \(x\), every ring is a discrete valuation ring with the same \(\pi\). The local transition maps are injective, since they take a nonzero \(u\pi^m\) to a nonzero element of that form. Their colimit is a domain, each nonzero element is still \(u\pi^m\), and each nonzero ideal is generated by an element of least valuation. Thus this colimit is a discrete valuation ring, including Noetherianity. Its fraction field is the colimit of the finite separable generic fields and is algebraic separable over \(K\). This degree-zero statement does not assert a higher-cohomology limit theorem.

AB.5. Local lifts of every divisor section

For a quasi-compact chart \(U\), its generic fibre consists of the generic fields of its components. Hence \(j_*\mathbf G_{m,K}(U)\) is the product of their multiplicative groups. A nonzero rational function on a component has finite divisor support: on a finite affine cover, numerator and denominator have zero-dimensional closed zero sets, which are finite in a Noetherian one-dimensional scheme. Its valuations give a section of the divisor sheaf, using the exact section description (S.2). AB.4 shows that these maps commute with restriction.

The kernel consists precisely of regular units. Valuation zero at every closed point puts both the function and its inverse in all the closed local rings. Their local representatives glue on the integral component to a regular function and its regular inverse; the generic point is already included. Thus the kernel agrees with \(\mathbf G_m\) on every chart.

To prove surjectivity as a sheaf map, let \(D=\sum_y n_y[y]\) be a section of (S.2), with finite support. At each support point choose a rational representative of a uniformizer \(\pi_y\). Shrink an open neighbourhood \(W_y\) to remove the other support points and all the other zeros and poles of \(\pi_y\). Then

On the complement of the support, the function \(1\) is a lift. These opens cover \(U\), and open immersions are étale. The local epimorphism criterion of T.1 proves surjectivity, establishing (R.14) directly. This construction does not require the divisor to be globally principal.

Finally, for a closed \(k\)-point \(x\), the sheaf \(i_{x*}A\) has sections \(A^{U_x}\), because AB.1 identifies the fibre as a discrete union of copies of \(\operatorname{Spec}k\). A short exact sequence of groups stays exact under this product: coordinates in a surjection can be lifted separately. Thus \(i_{x*}\) is exact. It preserves injectives by AB.3, and AB.2 makes sections over \(k\) exact. Applying it to an injective resolution gives \(H^q(X,i_{x*}A)=0\) for \(q>0\). Appendix S supplies the additional argument for the infinite point sum; exactness and acyclicity of its individual summands alone would not suffice.

Appendix AC. Why the curve site admits finite covers

Appendix S uses finite covering families on a basis stable under fibre products. The geometric input is that the covering maps are open. We prove the Noetherian finite-type image theorem and the flat prime-lifting argument, then verify flatness for the étale curve maps in question directly from their local parameters. This supplies the basis properties used in the divisor-sheaf calculation.

AC.1. A dominant image contains an open subset

Let \(A\subset B\) be an inclusion of domains, with \(B\) a finite-type \(A\)-algebra, and write \(K=\operatorname{Frac}A\). The normalization argument of K.1, extended to every field in Y.5, gives algebraically independent elements \(t_1,\ldots,t_d\) in \(B\otimes_AK\) such that this algebra is finite over \(K[t_1,\ldots,t_d]\). Invert a nonzero \(a\in A\) to put all the \(t_i\) in \(B_a\). Each member of a finite algebra generating set for \(B\) satisfies a monic equation over that polynomial ring. Invert one further product of denominators so that all coefficients belong to \(A_a[t_1,\ldots,t_d]\). Thus