Comparison with singular cohomology

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Prerequisites linked, Sections 10, 13 and 14 revised, and the orientation theorem of Section 10 proved, by Claude Opus 5.5 (Anthropic), October 2026. Trace-normalization arguments and their examples reconciled by GPT-6 Astra (OpenAI) in Codex, Ultra, October 2026. Original text released under CC0.

An étale neighbourhood of a complex point becomes a local analytic chart. This observation gives a map on cohomology, but it does not prove that the map is an isomorphism. We prove the compact-support comparison by reducing proper maps to projective curves, where coverings and line bundles determine the answer. We then prove ordinary comparison for constructible coefficients. Keeping these arguments separate also reveals a simple degree-zero counterexample when constructibility is removed.

1. Scope and the inputs that remain inputs

Schemes are of finite type over \(\mathbf C\). A morphism \(f:X\to S\) under consideration is separated and of finite type. The notation \(X^{\mathrm{an}}\) means the associated complex analytic space with its classical topology. An arbitrary torsion sheaf means an étale sheaf of abelian groups whose geometric stalks are torsion; it need not have a common annihilator. A constructible torsion sheaf has finite stalks on a finite stratification, and therefore has a common annihilator. We first work with \(\Lambda=\mathbf Z/n\), \(n\geq2\), and then pass to arbitrary torsion.

On a complex manifold, classical sheaf cohomology with constant coefficients is singular cohomology (Sheaf cohomology and singular cohomology on manifolds, Theorem 1.3). For a general constructible sheaf, the right side of the comparison is classical sheaf cohomology; one must retain its specialization maps. Ordinary singular cochains with a single coefficient group do not describe such a sheaf.

The lesson uses two analytic theorems proved in other courses:

• Riemann existence: analytification gives an equivalence between finite étale covers of any finite-type complex scheme and finite topological covering spaces of its analytic space, with no normality assumption (The Riemann existence theorem, Theorem 2.1). We use full faithfulness as well as existence; this includes group actions on covers.
• Projective-curve GAGA: for a projective complex curve, analytification is an equivalence between coherent algebraic and coherent analytic sheaves (Serre's comparison theorems and Chow's theorem, Theorems 4.1 and 5.2), compatible with tensor products. In particular it identifies algebraic and holomorphic line bundles, their morphisms and tensor products, and hence their Picard groups.

The scheme-theoretic prerequisites are finite normalization over \(\mathbf C\), the projectivity of a proper normal complex curve, and the invariance of the étale topos under nilpotent thickening. These are the same geometric foundations used in Lessons 6 and 12. Lesson 11 supplies constructible presentations, their Serre property, and filtered-cohomology continuity on a Noetherian étale site. Lessons 13–14 supply proper base change, compactification and \(Rf_!\), including their natural maps. General proper constructibility retains the stated-input status recorded in Lesson 14. Lessons 17–18 supply normalized curve and smooth duality; their exact current versions are prerequisites.

For the ordinary comparison argument in Section 11 we also use resolution of singularities: a reduced finite-type complex scheme has a proper map from a smooth scheme which is an isomorphism over a dense open (Resolution of singularities in characteristic zero, Corollary 5.1). The proper-support proof, Sections 5–9, does not use it.

The classical topological and analytic facts used are proved in the following lessons. Duality and dimension for the sheaf cohomology of manifolds, including compactly supported cohomology computed by extension by zero into a compact space, are Corollaries 5.1 and 5.2 and Theorem 4.1 of Sheaf cohomology and singular cohomology on manifolds; complex manifolds carry their canonical orientation by Orientations and fundamental classes, Corollary 3.5. A compact smooth manifold has a finite compatible triangulation by Smooth triangulations and the integral PL obstruction, Theorem N.1. The universal coefficient sequence is Thom classes and Euler classes, Theorem 5.2. Coherent cohomology of an \(n\)-dimensional complex manifold vanishes above degree \(n\) by The Dolbeault complex, Corollary 3.3. For the orientation theorem of Section 10 we use the orientation of products and of local biholomorphisms (Orientations and fundamental classes, Proposition 3.8 and the remark after Corollary 3.5), the integration map and its point classes (Poincaré duality, Corollary 2.5), and the class of the coordinate of the punctured plane under the Kummer sequence, with its sign (Sheaf cohomology and singular cohomology on manifolds, Corollary 6.4).

2. From analytic charts to a morphism of topoi

If \(u:U\to X\) is étale, the analytic inverse function theorem makes \(u^{\mathrm{an}}\) a local biholomorphism, hence a local homeomorphism. Surjective étale families give jointly surjective analytic families. Analytification preserves finite fibre products. The site of local homeomorphisms over a space presents its usual sheaf topos: an open subset is a local homeomorphism, and every local homeomorphism is covered by such opens.

Consequently the assignment \(U\mapsto U^{\mathrm{an}}\) induces a geometric morphism

One can see its inverse image concretely: form local analytic sections from étale sections and sheafify. At a complex point \(x\), the associated fibre functor is the geometric étale point \(\bar x:\operatorname{Spec}\mathbf C\to X\). Thus

Lemma 2.1. Analytification of abelian sheaves is exact, preserves colimits and tensor products, and sends finite locally constant sheaves to finite local systems. For an open immersion \(j\) and a closed immersion \(i\),

Proof. Inverse image of a geometric morphism preserves finite limits and all colimits. For abelian sheaves this proves exactness. Tensor products are constructed from sums and relations, so they are preserved; exactness also gives the derived tensor compatibility. A finite local system is trivialized by a finite étale cover, which analytifies to a covering. For (2.3), at a point of the open the stalk is the original stalk and outside it the stalk is zero. The closed case has the same stalk test. Formula (2.2) proves the assertions, including the canonical maps. \(\square\)

For finite-type complex schemes, closed complex points give all analytic points. To prove a morphism of analytic sheaves is an isomorphism, their stalks suffice. We will therefore apply algebraic proper base change only at these geometric points; no comparison over a non-complex residue field is being assumed.

3. Constructing the comparison maps

The square of topoi associated with \(f\) commutes. For a bounded-below complex \(E\), the counit \(f^*Rf_*E\to E\), pulled back to the analytic space and transposed by adjunction, defines

More explicitly, the map being transposed is

For the structure map to a point this gives the canonical ordinary map \(R\Gamma(X,E)\to R\Gamma(X^{\mathrm{an}},E^{\mathrm{an}})\). It is a map, without a prior assertion that it is an isomorphism.

Choose a relative compactification \(X\xrightarrow{j}\overline X\xrightarrow{p}S\), with \(j\) open and \(p\) proper, as in Lesson 14. Using (2.3) and (3.1) defines

Algebraic proper maps analytify to proper continuous maps. The associated analytic compactification therefore defines the usual derived direct image with proper support. A common proper refinement of two compactifications identifies their \(j_!\) objects by the fibre calculation proved in Section 7. Naturality of the counits in (3.1) then identifies their maps (3.2). Thus \(\beta_f\) is independent of compactification.

Lemma 3.1 (compatibilities). The comparison maps are natural in coefficients, respect distinguished triangles, and commute with composition, base-change maps, and the ordinary action on compactly supported cohomology. In particular they respect cup products.

Proof. The maps arise from inverse image, units and counits. Transposing two counits in sequence gives the counit of their composite, by the two adjunction identities. Transposing around a base-change square gives its usual base-change map by the same identities. Exact inverse image preserves the localization sequences (2.3), so the constructions also commute with their boundary maps. Finally inverse image preserves the evaluation \(E\otimes^LG\to H\); the projection maps for pushforward are obtained by transposing that evaluation. Pulling back either construction gives the same evaluation on the analytic side. Applying ordinary or proper-support sections proves cup compatibility, including mixed ordinary/compact products. The derived symmetry is the usual Koszul symmetry on both sides, so the sign for exchanging degrees \(a,b\) is \((-1)^{ab}\). \(\square\)

This supplies comparison maps between Leray and hypercohomology spectral sequences. Their finite filtrations, rather than just numerical dimensions, will be used below.

4. The classical proper-support tools

Here are the classical facts corresponding to the algebraic theorems already proved in the course.

Lemma 4.1 (compact continuity). On a compact Hausdorff space \(K\), sheaf cohomology commutes with filtered colimits of abelian sheaves.

Proof. First global sections commute. A section of the colimit has local representatives. Choose a finite cover by compact neighbourhoods contained in the representative opens, with interiors covering \(K\). On each compact intersection, equality of two representatives is locally witnessed at later stages, and finitely many such witnesses suffice. A common stage makes the representatives equal on all intersections; they glue on the interiors. The same finite-cover argument makes a section which becomes zero vanish at a single later stage. This proves both surjectivity and injectivity of the degree-zero colimit map. It applies to every closed subset of \(K\), which is also compact Hausdorff.

Use functorial injective resolutions of the system. Injective sheaves are flasque and hence soft on a compact Hausdorff space. A filtered colimit of these terms is soft: sections on each closed subset and global sections commute with the colimit, and the extension map was surjective at every stage. Soft sheaves on a paracompact Hausdorff space are acyclic for global sections, by the usual finite shrinking-and-extension proof of Čech acyclicity. The colimit resolution is exact on stalks and consists of these acyclic terms. Global sections of the resolution commute with the colimit, and filtered colimits of groups are exact. Its cohomology gives the assertion in every degree. \(\square\)

Lemma 4.2 (classical proper base change). If \(p:T\to B\) is proper between locally compact Hausdorff spaces, then

It follows that proper pushforward satisfies arbitrary base change.

Proof. The stalk on the left is cohomology over the inverse images of shrinking neighbourhoods of \(b\). These are cofinal among neighbourhoods of the compact fibre: if \(W\supset T_b\), then \(B\setminus p(T\setminus W)\) is an open neighbourhood whose inverse image lies in \(W\), since \(p\) is closed. Restriction to a compact Hausdorff closed subset computes the colimit of cohomology over its neighbourhoods. To verify the latter assertion, lift a finite Čech representative on the subset to neighbourhoods, shrink a finite cover so that its overlap equations hold, and do the same for a coboundary. Čech and derived cohomology agree on a compact Hausdorff space and the restrictions of injective sheaves are acyclic by this same refinement argument. This proves (4.1). After base change the two fibres are homeomorphic and carry the same sheaf; the stalk map is their identity, proving base change. \(\square\)

The full neighbourhood-refinement proof is in the pinned cohomology chapter cited in Section 17; Anschütz, Lemma 4.20 and Theorem 4.16, provide an additional open account. These statements apply to arbitrary abelian sheaves, without constructibility. From (4.1) and Lemma 4.1, \(R^qp_*\) commutes with filtered colimits for a proper analytic map. Applying this to \(p_*j_!\) gives the same assertion for proper-support pushforward in our compactifiable setting.

5. The comparison on a smooth projective curve

Let \(C\) be a connected smooth projective complex curve. Choose \(\zeta_n=\exp(2\pi i/n)\), identifying \(\Lambda(1)=\mu_n\) with the constant analytic \(\Lambda\). We keep twists when no choice is intended.

Theorem 5.1. The canonical maps

are isomorphisms in all degrees.

Proof in degree zero. Sections of a finite constant set are sections of its trivial finite cover. Full faithfulness in Riemann existence identifies those sections on the two sides and their group operations. In particular a connected algebraic \(C\) has connected analytic space: a nontrivial analytic open-and-closed decomposition would give a nonconstant section of the trivial two-sheeted cover. Hence both groups in degree zero are \(\Lambda\), and (5.1) is the identity on constants.

Proof in degree one. \(H^1(-,\Lambda)\) classifies \(\Lambda\)-torsors. The algebraic torsors are finite étale covers with a specified simply transitive fibre action. Riemann existence identifies the covers, their actions, and equivariant isomorphisms. The action conditions can be checked on fibres and are preserved. Analytification of a torsor represents precisely the cohomological map (5.1), since the Čech transition cocycle pulls back to the same cocycle. This proves an additive isomorphism, without assuming that higher cohomology is group cohomology of the fundamental group.

Proof in degree two. Kummer is exact both étale locally and analytically locally; a nonvanishing holomorphic function has a local holomorphic \(n\)-th root. Naturality gives a diagram of Kummer sequences. On the algebraic curve, Lesson 10 gives \(H^2(C,\mathbf G_m)=0\), so

On the Riemann surface, the exponential sequence

has \(H^2(C^{\mathrm{an}},\mathcal O_C)=0\), by The Dolbeault complex, Corollary 3.3, and \(H^3(C^{\mathrm{an}},\mathbf Z)=0\), by Sheaf cohomology and singular cohomology on manifolds, Corollary 5.1, since \(C^{\mathrm{an}}\) is a manifold of real dimension two. Its long exact sequence gives \(H^2(C^{\mathrm{an}},\mathcal O_C^*)=0\). Analytic Kummer therefore gives \(H^2(C^{\mathrm{an}},\mu_n)=\operatorname{Pic}(C^{\mathrm{an}})/n\). Curve GAGA identifies the two Picard groups. The map (5.1), transferred along \(\zeta_n\), is consequently an isomorphism in degree two. The algebraic groups vanish above two by Lesson 12; the analytic constant groups do by the same Corollary 5.1. This proves every degree. \(\square\)

The class of a point generates \(H^2(C,\mu_n)\), with trace \(1\) (Lesson 17). Its image under (5.1) is therefore a generator of \(H^2(C^{\mathrm{an}},\mu_n)\), which is free of rank one over \(\Lambda\). Theorem 10.1 identifies this image: under the trivialization by \(\zeta_n\) it is the positive topological point class of the complex orientation. So the curve trace of Lesson 17 is integration over the complex orientation when \(\mu_n\) is identified with \(\Lambda\) by \(\zeta_n\), and is minus integration when it is identified by \(\zeta_n^{-1}\).

6. Finite-cover resolutions of constructible sheaves

We need more than constant coefficients on a curve. The following embedding works in every dimension and will be used twice.

Lemma 6.1. For a reduced finite-type complex scheme \(X\) and a constructible \(\Lambda\)-sheaf \(F\), there is a monomorphism

where each \(\pi_a:T_a\to X\) is finite, \(T_a\) is normal, and \(\dim T_a\leq\dim X\). The cokernel is constructible. If \(X\) is proper of dimension at most one, the \(T_a\) are disjoint unions of smooth projective curves and points.

Proof. Choose a dense open \(U\) which is a disjoint union of normal opens of the irreducible components and on which \(F\) is finite locally constant. Write \(j:U\to X\) and \(i:Z=X\setminus U\to X\). The map

is injective: on \(U\) its first component is the identity, and at a point of \(Z\) its second component is the identity.

A finite local system on a connected normal \(U\) is trivialized by a finite étale cover \(r:V\to U\). Its finite fibre module \(M\) embeds in \(\Lambda^r\). To see this even for composite \(n\), decompose \(M\) into cyclic groups \(\mathbf Z/d\), \(d\mid n\), and embed \(1\) as \(n/d\) in a separate copy of \(\Lambda\). Equivalently, \(\Lambda\) is a finite injective cogenerator, as proved in Lesson 17. The unit \(F|_U\to r_*r^*F|_U\), followed by this embedding on the trivialized fibres, is injective and lands in \(r_*\Lambda_V^r\).

Normalize each component of \(X\) in the finitely many function fields of \(V\). Excellence makes the resulting map \(\pi:T\to X\) finite, and its restriction over \(U\) is \(V\). On a normal scheme \(T\), with dense open \(j':V\to T\),

Indeed a strict local normal connected scheme is integral; its intersection with the dense open is nonempty and irreducible, hence connected. Sections of the constant sheaf there are exactly \(\Lambda\). These are the geometric stalks of (6.3). Applying left exact \(j_*\) to the embedding on \(U\) now embeds \(j_*F|_U\) in \(\pi_*\Lambda_T^r\).

For the boundary term in (6.2), use the same construction by induction on \(\dim Z<\dim X\), beginning with finite sets of points. Composing its finite maps with the closed immersion \(i\) gives the remaining terms of (6.1). All terms and the cokernel are constructible by the constructible Serre property and finite pushforward. A proper normal complex curve is smooth and projective; zero-dimensional normal schemes are finite sets of complex points. This proves the final assertion. \(\square\)

Iterating (6.1) on its cokernel produces an exact right resolution \(0\to F\to I^0\to I^1\to\cdots\). It need not terminate. Both analytification and finite pushforward are exact; finite pushforward commutes with analytification by proper base change with zero-dimensional fibres. The hypercohomology spectral sequence is

It is first quadrant: each fixed total degree involves only finitely many pairs \((a,b)\). Thus if comparison is an isomorphism on every \(I^a\), it is an isomorphism on \(F\); there is no claim that an infinite resolution is an acyclic resolution. The same conclusion follows by truncating it beyond the fixed degree and noting the remaining cokernel shifts past that degree.

For a proper curve, Theorem 5.1, finite pushforward and (6.4) prove comparison for every constructible \(F\). Points contribute only degree zero. This includes singular and reducible curves.

7. Proper comparison by lowering the dimension

Lemma 7.1 (proper modification over an open). Suppose \(h:Y\to X\) is proper and is an isomorphism over \(U\subset X\). For \(j:U\to X\), \(j':h^{-1}U\to Y\), and any torsion \(G\) on \(U\), the unit gives

The same statement holds analytically, and comparison commutes with these identifications.

Proof. At a geometric point of \(U\), the proper fibre is a single point and the map is the identity on its coefficient. At a point outside \(U\), the restriction of \(j'_!G\) to the entire fibre is zero. Algebraic proper base change therefore makes (7.1) an isomorphism on every stalk. Classical proper base change gives the same conclusion analytically. Both maps are units, so their comparison squares commute by Section 3. \(\square\)

Theorem 7.2. Proper comparison (3.1) is an isomorphism for every proper finite-type complex map and every constructible torsion sheaf.

Proof. Induct on a bound \(d\) for the dimensions of the proper fibres. For \(d=0\) the map is finite, and its stalk is a finite direct sum over points; this proves the assertion, including the vanishing of higher images. For \(d=1\), proper base change on both sides reduces the assertion at each analytic point of the base to a proper complex curve or a finite set of points. Section 6 proves that case.

Assume \(d\geq2\) and the result for smaller relative dimensions. Again proper base change reduces us to a proper scheme \(X/\mathbf C\) of dimension at most \(d\). Reduce \(X\) without changing its étale topos or underlying analytic topology. Zero-dimensional components separated from the others can be handled separately; on the remaining positive-dimensional components choose a rational function nonconstant on each component. On a dense open \(U\), where the components are disjoint, it defines \(\varphi:U\to\mathbf A^1\). Let \(Y\) be the reduced closure of its graph in \(X\times\mathbf P^1\). There are proper maps

Every irreducible component of \(Y\) dominates \(\mathbf P^1\), since the function was nonconstant. Its fibres are proper closed subsets of that component, so have dimension at most \(d-1\). The fibres of \(b\) have dimension one, also less than \(d\).

Put \(G=F|_U\) and \(E=j'_!G\). The induction hypothesis compares \(Ra_*E\) with \(Ra^{\mathrm{an}}_*E^{\mathrm{an}}\). Its cohomology sheaves are constructible by proper constructibility and vanish above the uniform fibre bound. The already known comparison for \(b\), applied to these sheaves and then their finite truncations, compares the two Leray spectral sequences. Their finite filtrations in each total degree give

Lemma 7.1 transfers (7.3) to \(R\Gamma(X,j_!G)\). For the closed complement \(Z=X\setminus U\), every component has dimension at most \(d-1\). The induction hypothesis compares \(R\Gamma(Z,i^*F)\). The exact localization sequence

and its analytic counterpart now compare \(R\Gamma(X,F)\). Finally return to the original proper map: at each analytic point the comparison stalk is this comparison on its complex fibre. Stalks detect the desired isomorphism. This completes the induction. \(\square\)

Notice the two dimensions in this argument. It lowers proper fibre dimension through \(a\), and lowers support dimension on the boundary \(Z\). Merely choosing a curve inside \(X\) would accomplish neither reduction.

8. Arbitrary torsion in the proper theorem

Theorem 8.1. For proper \(p:X\to S\) and an arbitrary torsion sheaf \(F\),

Proof. On the Noetherian étale site of \(X\), \(F\) is a filtered colimit of constructible finite torsion sheaves, using their finite presentations as in Lesson 11. One may first write \(F=\varinjlim_nF[n]\), with divisibility indexing the annihilators, then use the finite presentations for each \(\mathbf Z/n\)-sheaf. The coherent-site continuity theorem makes \(R^qp_*\) commute with these filtered colimits. Inverse image and analytification commute with colimits. On the analytic side classical proper base change computes each stalk on the compact proper fibre, and Lemma 4.1 proves its continuity. Hence (8.1) is the filtered colimit in each degree of the isomorphisms in Theorem 7.2. Since the comparison maps themselves are natural in \(F\), the resulting isomorphism is the original map (3.1). \(\square\)

This passage does not replace an arbitrary torsion sheaf by one finite local system or assume a common integer kills it. Properness supplies the compactness on the analytic side that makes the passage legitimate.

9. Artin comparison with proper support

Theorem 9.1. For separated finite-type \(f:X\to S\), arbitrary torsion \(F\), and every \(q\geq0\), the canonical map is an isomorphism:

In particular, for a separated finite-type complex scheme,

Proof. Use the compactification in (3.2). Theorem 8.1 applies to the arbitrary torsion sheaf \(j_!F\) on \(\overline X\), and Lemma 2.1 identifies its analytification with \(j^{\mathrm{an}}_!F^{\mathrm{an}}\). This is exactly (3.2), hence is an isomorphism. Taking cohomology sheaves proves (9.1); taking the base to be a point proves (9.2). If \(S\) is not separated, work over its affine open charts. The inverse image \(X_V\) over such a chart is separated over \(\mathbf C\), so the analytic proper-support argument applies there. The canonical maps agree on overlaps by naturality, and their isomorphisms glue. \(\square\)

For bounded constructible complexes the same theorem follows by finite truncations. Composition and all base-change diagrams retain the compatibilities of Section 3. The proof has used precisely the projective-curve comparison, finite-cover coefficient resolutions, proper fibre calculations, the graph reduction and continuity. It has not used an ordinary comparison theorem or a resolution of higher-dimensional singularities.

10. Orientation, the top degree and smooth ordinary comparison

The cup-product compatibility was proved in Lemma 3.1 before any isomorphism assertion. This section first compares the étale trace with integration over the complex orientation, and then proves smooth ordinary comparison from cup products, the compact comparison of Theorem 9.1, étale duality from Lesson 18, and duality for the sheaf cohomology of the complex manifold \(X^{\mathrm{an}}\). Throughout, classical compactly supported cohomology is computed as in Section 3, by extension by zero into the analytic space of a compactification \(X\subset\overline X\); this space is compact Hausdorff, since \(\overline X\) is proper. By Sheaf cohomology and singular cohomology on manifolds, Theorem 4.1 and Corollary 3.2 it is the compactly supported singular cohomology of \(X^{\mathrm{an}}\), and by Theorem 2.2 there, cohomology supported at a closed point \(x\) is \(H^*(X^{\mathrm{an}},X^{\mathrm{an}}\setminus x)\).

For a closed point \(x\) of a smooth \(X\) of pure dimension \(d\), Lesson 18, Section 12, gives the point class \(\operatorname{cl}_x\in H^{2d}_x(X,\Lambda(d))\), the purity generator, and its compact image \(\operatorname{cl}_c(x)\), with \(\operatorname{Tr}_X(\operatorname{cl}_c(x))=1\) by (12.3). On the analytic side, \(u_x\in H^{2d}(X^{\mathrm{an}},X^{\mathrm{an}}\setminus x;\Lambda)\) and \([x]\in H^{2d}_c(X^{\mathrm{an}};\Lambda)\) denote the topological point classes of the complex orientation, and \(\int_{X^{\mathrm{an}}}\) the integration map, as in Poincaré duality, Corollary 2.5.

Theorem 10.1 (orientation). Let \(X\) be smooth, separated and of finite type over \(\mathbf C\), of pure dimension \(d\), and identify \(\Lambda(1)\) with \(\Lambda\) by \(\zeta_n=\exp(2\pi i/n)\).

1. For every closed point \(x\), comparison sends \(\operatorname{cl}_x\) to \(u_x\), and compact comparison sends \(\operatorname{cl}_c(x)\) to \([x]\).
2. For every \(a\in H^{2d}_c(X,\Lambda(d))\), \[ \operatorname{Tr}_X(a)=\int_{X^{\mathrm{an}}}c_c(a). \tag{10.1} \]
3. Let \(f:X\to S\) be smooth, separated, of finite type and of constant relative dimension \(d\), with \(S\) of finite type over \(\mathbf C\). At every \(s\in S(\mathbf C)\), after the base-change identifications of the stalks of \(R^{2d}f_!\Lambda(d)\) and \(R^{2d}f^{\mathrm{an}}_!\Lambda\) with the top compactly supported groups of the fibre (Lesson 18, property (T1), and Lemma 4.2), the stalk of \(\operatorname{Tr}_f\) is integration over the complex orientation of \(X_s^{\mathrm{an}}\).

If \(\Lambda(1)\) is identified with \(\Lambda\) by \(\zeta_n^{-1}\) instead, each twist contributes \(-1\). The images in (1) acquire \((-1)^d\), and the traces in (2) and (3) are \((-1)^d\) times integration. These are two trivializations of the same twisted comparison.

Proof. The last sentence holds because the two identifications of \(\Lambda(d)\) with \(\Lambda\) differ by \((-1)^d\).

Step 1: the origin of the affine line. Let \(t\) be the coordinate of \(\mathbf A^1\) and \(\kappa(t)\in H^1(\mathbf A^1\setminus\{0\},\mu_n)\) its Kummer class. Lesson 17, Lemmas 4.2–4.3, identify the trace-positive purity generator with the negative localization boundary of the Kummer class of a uniformizer. Their cochain calculation retains the divisor convention \(\partial_{\mathrm{div}}[x]=[\mathcal O(-x)]\) and proves that this negative boundary maps to \(c_1^{(n)}\mathcal O(x)\). Lesson 18 uses exactly this orientation (proof of Lemma 12.2). The localization sequence of \(\mathbf A^1\) maps to the one of the strict henselization at \(0\) used there, and \(H^2_{\{0\}}(\mathbf A^1,\mu_n)\) is the closed stalk of \(R^2i^!\mu_n\); hence \(\operatorname{cl}_0=-\partial\kappa(t)\), with \(\partial\) the localization boundary. By Lemma 3.1, comparison commutes with the Kummer and localization boundaries, so it sends \(\operatorname{cl}_0\) to \(-\partial\kappa(z)\), where \(z\) is the coordinate of \(\mathbf C\). By Sheaf cohomology and singular cohomology on manifolds, Corollary 6.4(2), \(\langle\partial\kappa(z),o_2\rangle=\zeta_n^{-1}\), and \(o_2\) is the complex orientation of \(\mathbf C\) at \(0\). Negating the supported Kummer class inverts its value on \(o_2\) from \(\zeta_n^{-1}\) to \(\zeta_n\). Under the identification by \(\zeta_n\), comparison therefore sends \(\operatorname{cl}_0\) to \(u_0\).

Step 2: the point \(p=(0,\ldots,0)\) of \(P=(\mathbf P^1)^d\). Groups supported at \(0\) are the same for \(\mathbf A^1\) and \(\mathbf P^1\), compatibly with purity and with the analytic groups (sections supported in a closed subset of an open set extend by zero). Let \(h\in H^2(\mathbf P^1,\Lambda(1))\) be the image of \(\operatorname{cl}_0\), so \(\operatorname{Tr}(h)=1\) by (12.3), and put \(h_i=\operatorname{pr}_i^*h\) on \(P\). Since \(P\) is proper, compact and ordinary cohomology agree. Lesson 18, Lemma 13.2, applied repeatedly, gives \(\operatorname{Tr}_P(h_1\cup\cdots\cup h_d)=1\); all degrees are even, so no sign occurs. The trace is an isomorphism \(H^{2d}(P,\Lambda(d))\to\Lambda\) (Lesson 18, property (T5)) and \(\operatorname{Tr}_P(\operatorname{cl}_c(p))=1\), so \(\operatorname{cl}_c(p)=h_1\cup\cdots\cup h_d\). By Lemma 3.1 and Step 1, comparison sends this class to \(\operatorname{pr}_1^*[0]\cup\cdots\cup\operatorname{pr}_d^*[0]=[0]\times\cdots\times[0]\), where \([0]\in H^2(\mathbf{CP}^1;\Lambda)\) is the point class. The complex orientation of \((\mathbf{CP}^1)^d\) is the product orientation Orientations and fundamental classes, Proposition 3.8, so \(\int[0]\times\cdots\times[0]=1\) by Poincaré duality, Corollary 2.5(2) and (5), and \([0]\times\cdots\times[0]=[p]\) by Corollary 2.5(1) there. Thus compact comparison sends \(\operatorname{cl}_c(p)\) to \([p]\). The maps from cohomology supported at \(p\) to \(H^{2d}(P,\Lambda(d))\) and to \(H^{2d}(P^{\mathrm{an}};\Lambda)\) are isomorphisms, since on each side both groups are free of rank one over \(\Lambda\) and the point class goes to a generator, and they commute with comparison. So comparison sends \(\operatorname{cl}_p\) to \(u_p\).

Step 3: any closed point. Let \(x\in X\) be closed. By AI Integrated Stacks Project, Lemma 054L there are an open \(U\ni x\) and an étale \(g:U\to\mathbf A^d\); after a translation \(g(x)=0\), and after removing the finitely many other points of \(g^{-1}(0)\), \(g^{-1}(0)=\{x\}\). Purity commutes with restriction to open subsets and with étale localization (Lesson 18, the remark after Lemma 12.2, resting on (11.5) and property (T2)). Hence restriction from \(X\) to \(U\) sends \(\operatorname{cl}_x\) to the point class of \(x\) in \(U\), which is \(g^*\) of the point class of \(0\) in \(\mathbf A^d\), which is the restriction of \(\operatorname{cl}_p\). On the analytic side, restriction to open sets preserves the topological point classes, by excision. The map \(g^{\mathrm{an}}\) is a local biholomorphism at \(x\), so it preserves the complex orientation at \(x\) (Orientations and fundamental classes, the remark after Corollary 3.5), and it maps \(U^{\mathrm{an}}\setminus x\) into \(\mathbf C^d\setminus0\); by Poincaré duality, Corollary 2.5(4), applied on a neighbourhood of \(x\) mapped homeomorphically onto a neighbourhood of \(0\), \((g^{\mathrm{an}})^*u_0=u_x\). Comparison commutes with restrictions and with \(g^*\) (Lemma 3.1), and restriction from \(X\) to \(U\) is injective on the analytic groups supported at \(x\), by excision. Step 2 therefore gives the first assertion of (1). Comparison also commutes with the map from cohomology supported at \(x\) to compactly supported cohomology, which is induced by the counit of the closed immersion of \(x\) (Lemma 3.1); this gives the second.

Step 4: traces and families. For connected \(X\), the trace is an isomorphism \(H^{2d}_c(X,\Lambda(d))\to\Lambda\) (Lesson 18, property (T5)) with \(\operatorname{Tr}_X(\operatorname{cl}_c(x))=1\), and \(\int_{X^{\mathrm{an}}}\) is an isomorphism with \(\int[x]=1\) (Poincaré duality, Corollary 2.5(1) and (2)), since \(X^{\mathrm{an}}\) is connected (Connectedness and full faithfulness, Theorem 2.1). By (1) the two sides of (10.1) agree on the generator \(\operatorname{cl}_c(x)\), hence everywhere; for disconnected \(X\), sum over the finitely many components. For (3), property (T1) identifies the stalk of \(\operatorname{Tr}_f\) at \(s\) with \(\operatorname{Tr}_{X_s}\), the comparison maps commute with base change (Lemma 3.1), and Lemma 4.2 computes the analytic stalk; apply (2) to \(X_s\). \(\square\)

Two signs enter the one-dimensional calculation: continuing an \(n\)-th root of \(z\) once around the counterclockwise circle multiplies it by \(\zeta_n\), while the connecting map of the Kummer sequence pairs with that circle to \(\zeta_n^{-1}\) (Sheaf cohomology and singular cohomology on manifolds, Proposition 6.3 and Corollary 6.4). The trace-positive purity generator is the negative of the raw supported Kummer class. These two signs cancel, so the degree trace agrees with complex-orientation integration under the positive-root trivialization.

Theorem 10.2 (smooth constant ordinary comparison). If \(X\) is smooth, separated and finite type over \(\mathbf C\), then

for every \(r\).

Proof. Work on each connected component, of dimension \(d\), and write \(c\) and \(c_c\) for the ordinary and compact comparison maps. The compactly supported étale groups are finite, and \(c_c\) is an isomorphism by Theorem 9.1.

Injectivity. Étale duality in Lesson 18 identifies \(H^r(X,\Lambda)\) with the \(\Lambda\)-dual of \(H^{2d-r}_c(X,\Lambda(d))\), by the compact-first trace pairing. So for \(a\neq0\) in \(H^r(X,\Lambda)\) there is \(b\in H^{2d-r}_c(X,\Lambda(d))\) whose product with \(a\) is nonzero in \(H^{2d}_c(X,\Lambda(d))\). By Lemma 3.1, \(c_c\) maps this product to the product of \(c(a)\) and \(c_c(b)\), and \(c_c\) is injective; hence \(c(a)\neq0\).

Equal orders. Étale duality shows that \(H^r(X,\Lambda)\) is finite of the same order as \(H^{2d-r}_c(X,\Lambda(d))\), which by Theorem 9.1 is the order of \(H^{2d-r}_c(X^{\mathrm{an}},\Lambda)\), the twist being trivialized analytically by \(\zeta_n\). By Sheaf cohomology and singular cohomology on manifolds, Corollary 5.2(3) and (4), applied to the canonically oriented complex manifold \(X^{\mathrm{an}}\) and the analytic space of a compactification, \(H^r(X^{\mathrm{an}},\Lambda)\) is finite of that same order.

An injective map between finite groups of the same order is bijective. \(\square\)

Classical duality enters only through the equality of orders. It holds for every \(n\), because \(\mathbf Z/n\) is injective as a module over itself; duality over a field would cover only prime \(n\).

A second proof. With Theorem 10.1, Lemma 3.1 gives a commutative square from the étale pairing \(H^{2d-r}_c(X,\Lambda(d))\times H^r(X,\Lambda)\to\Lambda\), cup product followed by trace, to the classical pairing, cup product followed by integration. Both pairings are perfect: the first by Lesson 18, the second by Sheaf cohomology and singular cohomology on manifolds, Corollary 5.2(1) together with Poincaré duality, Corollary 2.3(2). Since \(c_c\) is an isomorphism, the square identifies \(c\) with the dual of \(c_c^{-1}\), so \(c\) is an isomorphism.

11. Ordinary comparison for all constructible coefficients

Theorem 11.1. For separated finite-type complex \(X\), constructible torsion \(F\), and every \(q\),

The same holds for bounded constructible complexes.

Proof. We induct on \(d=\dim X\), allowing all finite annihilators at each stage. Dimension zero is a finite set of points after reduction, so is immediate. Nilpotents can be removed without changing either cohomology theory.

First prove the assertion for \(\Lambda_X\) on every reduced \(X\) of dimension at most \(d\). By resolution of singularities (Section 1) there is a proper \(p:Y\to X\), with \(Y\) smooth, which is an isomorphism over a dense open \(U\) meeting every component. Put

Proper constructibility and the finite fibre cohomological bound make \(C\) a bounded constructible complex. Its restriction to \(U\) is zero, since the displayed map is the identity there. Thus it is supported on \(Z=X\setminus U\), of dimension less than \(d\). If \(i:Z\to X\), the unit \(C\to i_*i^*C\) is an isomorphism on stalks; hence the induction hypothesis, extended by finite truncations, compares its ordinary cohomology.

Proper comparison identifies \(C^{\mathrm{an}}\) with the cone of \(\Lambda_{X^{\mathrm{an}}}\to Rp^{\mathrm{an}}_*\Lambda_{Y^{\mathrm{an}}}\). Smooth ordinary comparison, Theorem 10.2, compares \(R\Gamma(Y,\Lambda)\); the derived identity \(R\Gamma(X,Rp_*\Lambda)=R\Gamma(Y,\Lambda)\) and its analytic version are natural. The map between the two triangles defined by (11.2) now has isomorphisms on the second and third terms. The long exact sequences prove that it is an isomorphism on \(R\Gamma(X,\Lambda)\) as well. This proves constant comparison in dimension \(d\), for every \(n\), without requiring \(X\) to be proper or smooth.

Now take an arbitrary constructible \(\Lambda\)-sheaf \(F\) on such an \(X\). Lemma 6.1 provides its right resolution by finite pushforwards of constant sheaves on normal finite-type schemes \(T_a\) of dimension at most \(d\). Constant comparison on each \(T_a\) has just been proved. Finite pushforward is exact and compares with its analytic counterpart, so the ordinary cohomology of every term \(I^a\) compares. The first-quadrant spectral sequences (6.4) prove (11.1). This also proves the induction claim for the next dimension. Every constructible torsion sheaf has some annihilator \(n\), so nothing is lost by the coefficient notation. Bounded constructible complexes follow by their finite cohomology-sheaf truncation triangles. \(\square\)

This identifies the substantive extra work for ordinary comparison: smooth duality and a proper resolution whose error cone lives on a smaller-dimensional closed set. Proper base change alone does not compute stalks of \(Rj_*\) for a nonproper open immersion.

12. Why ordinary comparison needs constructibility

Let \(X=\mathbf A^1_{\mathbf C}\), let \(i_m:\{m\}\to X\) for \(m\in\mathbf Z\), and set

This is a torsion sheaf annihilated by \(n\), but is not constructible. The sum is the filtered colimit of finite sums. On the quasi-compact Noetherian étale site, global sections commute with that filtered colimit, by Lesson 11. Consequently

Analytification preserves the sum. In the complex plane the integers form a closed discrete subset, and every compact set meets only finitely many of them. Any family \((a_m)_{m\in\mathbf Z}\) is locally a section of a finite sum of the skyscrapers: near any point choose a neighbourhood meeting only finitely many integers. These sections agree on overlaps and glue. Conversely a section is determined by these stalk values. Thus

The canonical ordinary comparison is the inclusion of the direct sum in the product, and is not surjective. A family with value \(1\) at every integer exhibits the failure. Compactly supported sections on either side have finite support, so both \(H^0_c\) groups are the direct sum and their comparison is an isomorphism, as Theorem 9.1 requires.

The analytic direct sum of skyscraper sheaves has product global sections here because infinite families are locally finite. Confusing a sheaf direct sum with its presheaf of finite-support global sections would erase the example.

13. Projective spaces, an elliptic curve and the punctured line

Example 13.1 (projective space). The analytic space of \(\mathbf P^N_{\mathbf C}\) is \(\mathbf{CP}^N\). Lesson 18, Section 14, computes the étale cohomology ring: with \(h=c_1^{(n)}(\mathcal O(1))\),

the odd groups vanish, and \(\operatorname{Tr}(h^N)=1\). By Theorem 10.2 and Lemma 3.1, comparison is an isomorphism of graded rings onto the classical sheaf cohomology ring of \(\mathbf{CP}^N\), which is therefore \(\Lambda[h^{\mathrm{an}}]/\bigl((h^{\mathrm{an}})^{N+1}\bigr)\), with \(h^{\mathrm{an}}\) the image of \(h\). Restriction \(H^2(\mathbf{CP}^N;\Lambda)\to H^2(\mathbf{CP}^1;\Lambda)\) to a line is an isomorphism, being the analytic image of the algebraic restriction, an isomorphism by Lesson 18, Example 14.2. Let \([H]^+\in H^2(\mathbf{CP}^N;\Lambda)\) be the class whose restriction to a line is the point class of the line, the positive hyperplane class. Then \(h^{\mathrm{an}}=[H]^+\) when twists are trivialized by \(\zeta_n\): on a line \(L\) meeting a hyperplane transversally in one point \(q\), \(h\) restricts to the Gysin class of \(q\) (Lesson 18, Lemma 12.2 and the transverse-pullback remark after it), which Theorem 10.1 sends to \([q]\). Since \(\operatorname{Tr}(h^N)=1\), \(h^N\) is the point class, and Theorem 10.1 gives \((h^{\mathrm{an}})^N=[\mathrm{pt}]\). Hence \(([H]^+)^N=[\mathrm{pt}]\): the classical value \(\int_{\mathbf{CP}^N}([H]^+)^N=1\) follows from the étale one. If twists are suppressed using \(\zeta_n\), each even group is \(\Lambda\); without that choice the untwisted algebraic group is canonically \(\Lambda(-i)\).

Example 13.2 (elliptic curve). For an elliptic curve \(E\), Lesson 17 gives

and Poincaré duality for curves (Lesson 17, Theorem 11.1) makes the cup product pairing \(H^1(E,\Lambda)\times H^1(E,\Lambda(1))\to H^2(E,\Lambda(1))\cong\Lambda\) perfect. Theorem 5.1 transports these groups to the analytic torus \(E^{\mathrm{an}}\), and Lemma 3.1 transports the pairing; so the classical cup product pairing on \(H^1(E^{\mathrm{an}},\Lambda)\) is perfect as well. Riemann existence also identifies \(\pi_1^{\mathrm{\acute et}}(E)\) with the profinite completion of \(\pi_1^{\mathrm{top}}(E^{\mathrm{an}})\) (The Riemann existence theorem, Theorem 3.2). This calculation is compatible with the Jacobian description in Lesson 17; it does not require another proof of its separately stated Weil-pairing comparison.

Integrally, the cup product pairing on \(H^1(E^{\mathrm{an}};\mathbf Z)\) is unimodular and alternating. Indeed \(E^{\mathrm{an}}\) is a closed connected surface with its complex orientation, so \(H^2(E^{\mathrm{an}};\mathbf Z)\congH_0\cong\mathbf Z\) by Poincaré duality; its homology is finitely generated (it has a finite triangulation, Section 14), and \(H_1\) is torsion-free, since by universal coefficients its torsion would appear in \(H^2\). So \(H^1(E^{\mathrm{an}};\mathbf Z)=\operatorname{Hom}(H_1,\mathbf Z)\) is free, of rank two by (13.2) and Theorem 5.1. The pairing \((a,b)\mapsto\int a\cup b\) equals \(\langle b,[E^{\mathrm{an}}]\frown a\rangle\) (Cap products and cohomology with compact supports, Proposition 1.1(2)), and capping with the fundamental class is an isomorphism \(H^1\to H_1\), so the pairing is unimodular. It is alternating because \(a\cup a=-a\cup a\) in the torsion-free group \(H^2\). In a basis its matrix is therefore \(\pm\begin{pmatrix}0&1\\-1&0\end{pmatrix}\), and after ordering the basis suitably we obtain \(\alpha,\beta\) with \(\int\alpha\cup\beta=1\), \(\int\beta\cup\alpha=-1\) and \(\alpha^2=\beta^2=0\). The last equalities also hold modulo \(n\) for even \(n\), as reductions of integral classes of square zero, not as a consequence of skew symmetry alone. Let \(\alpha_{\mathrm{\acute et}},\beta_{\mathrm{\acute et}}\in H^1(E,\Lambda)\) be the classes whose images are the reductions of \(\alpha,\beta\) (Theorem 5.1). With twists trivialized by \(\zeta_n\), Theorem 10.1 gives \(\operatorname{Tr}(\alpha_{\mathrm{\acute et}}\cup\beta_{\mathrm{\acute et}})=1\), \(\operatorname{Tr}(\beta_{\mathrm{\acute et}}\cup\alpha_{\mathrm{\acute et}})=-1\) and \(\alpha_{\mathrm{\acute et}}^2=\beta_{\mathrm{\acute et}}^2=0\); with \(\zeta_n^{-1}\) the two traces are \(-1\) and \(1\).

Example 13.3 (\(\mathbf C^*\)). The analytic punctured line deformation retracts onto the unit circle, with positive counterclockwise generator \(\gamma\) of \(\pi_1^{\mathrm{top}}=\mathbf Z\). Riemann existence and the pointed fibre functors give

Indeed finite covers of the circle correspond to finite sets with a permutation, equivalently finite continuous \(\widehat{\mathbf Z}\)-sets; equivalence of their fibre functors identifies the automorphism groups. The connected cover \(z\mapsto z^r\) corresponds to the subgroup \(r\mathbf Z\), whose closures form a cofinal family of open subgroups. The Kummer class of \(t\) in \(H^1(\mathbf G_m,\mu_n)\) sends \(\gamma\) to \(\zeta_n\), since continuation of \(t^{1/n}\) once around the circle multiplies it by that root. Thus the comparison fixes both the generator and its orientation. This value is the monodromy of the torsor of \(n\)-th roots. The singular description of \(H^1\), through Sheaf cohomology and singular cohomology on manifolds, Theorem 1.3 and the Kronecker pairing, gives the inverse value: the Kummer class pairs with \([\gamma]\) to \(\zeta_n^{-1}\), because the connecting map is the negative of the monodromy (Proposition 6.3 and Corollary 6.4 there). The nonzero ordinary constant groups are \(H^0=\Lambda\), \(H^1=\Lambda(-1)\); the compact groups are \(H^1_c=\Lambda\), \(H^2_c=\Lambda(-1)\), in accordance with curve duality. These two different degree-one orientations should not be silently conflated.

14. Betti numbers and the coefficient limit

Let \(X\) be smooth projective over \(\mathbf C\), and \(\ell\) a prime. Write

Its analytic space is a compact complex manifold, which has a finite compatible triangulation (Smooth triangulations and the integral PL obstruction, Theorem N.1). Let \(C_\bullet\) be the simplicial chain complex of such a triangulation, a bounded complex of finitely generated free abelian groups, and \(C^\bullet=\operatorname{Hom}(C_\bullet,\mathbf Z)\), so that \(C^\bullet\otimes A=\operatorname{Hom}(C_\bullet,A)\) for every abelian group \(A\). Simplicial and singular homology agree [Fomberg, Proposition 1.32, in the core course Algebraic Topology]; applying the natural universal coefficient sequences (Thom classes and Euler classes, Theorem 5.2) and the five lemma to the comparison chain map, \(H^q(C^\bullet\otimes A)\) is singular cohomology with coefficients in \(A\), naturally in \(A\), and singular cohomology is sheaf cohomology by Sheaf cohomology and singular cohomology on manifolds, Theorem 1.3. For every \(m\), comparison therefore naturally identifies the algebraic \(\mathbf Z/\ell^m\)-cohomology with \(H^q(C^\bullet\otimes\mathbf Z/\ell^m)\); these identifications commute with reduction of coefficients.

Define the integral étale coefficient-limit complex as \(R\varprojlim_m R\Gamma(X,\mathbf Z/\ell^m)\). Each finite-coefficient cohomology group is finite, so its tower is Mittag–Leffler: for a fixed target the descending images stabilize. Its \(\varprojlim^1\) vanishes, and the Milnor sequence identifies the limit cohomology with the inverse limit of the cohomology groups. On the cellular side the degreewise reduction maps are surjective, so derived inverse limit is the ordinary complex \(C^\bullet\otimes\mathbf Z_\ell\). Consequently

Tensoring with the flat localization \(\mathbf Q_\ell\) gives

The finite complex justifies each passage; this is not an application of the torsion theorem directly to the nontorsion sheaf \(\mathbf Q_\ell\). Nor does (14.2) say that \(\dim_{\mathbf F_\ell}H^q(X,\mathbf F_\ell)=b_q\) always. The integral universal coefficient sequence has contributions from \(\ell\)-torsion in integral cohomology in degrees \(q\) and \(q+1\). The rational Betti number is the rank, with those contributions removed.

15. Exercises

1. Easy. Compute both cohomologies of \(\mathbf P^1_{\mathbf C}\) with \(\Lambda\)-coefficients, and identify the canonical comparison in every degree, including the sign of the image of its top generator.
2. Medium. Prove equality of étale and topological Betti numbers for a smooth projective complex variety. Explain why finite-coefficient dimensions need not equal rational Betti numbers.
3. Medium. Prove that ordinary and compact comparison preserve cup products, and that for a smooth separated \(f:X\to S\) of relative dimension \(d\) the relative trace corresponds on every fibre to integration over the complex orientation of the fibre, up to a sign that you determine. Specify the coefficient twists and the sign when two classes are exchanged.
4. Hard. Starting from Riemann existence and projective-curve Picard GAGA, carry out the reduction of the arbitrary-torsion proper-support comparison to smooth projective curves. Identify every dimension induction, coefficient resolution and limit argument. Then give the extra argument for ordinary constructible comparison and identify its additional geometric prerequisite.

16. Complete solutions

Solution 1. Algebraically, Kummer, \(\operatorname{Pic}(\mathbf P^1)=\mathbf Z\), and the curve computation give \(H^0=\Lambda\), \(H^1=0\), \(H^2=\Lambda(-1)\), with all higher groups zero. By Theorem 5.1 the analytic groups are the same after choosing \(\zeta_n\). The degree-zero comparison is the identity on constants. Degree one and all degrees above two compare zero groups. In degree two, naturality of Kummer and curve GAGA send \(c_1^{(n)}(\mathcal O(1))\) to its analytic Chern class modulo \(n\). The algebraic class is the class of a point, with trace \(1\) (Lesson 18, Lemma 12.2), so by Theorem 10.1 its image is the positive topological point class when twists are trivialized by \(\zeta_n\): it generates \(H^2(\mathbf{CP}^1,\Lambda(1))\), and its integral over the complex orientation is \(1\), equal to its trace. With \(\zeta_n^{-1}\) the integral is \(-1\). Without the chosen root this is a comparison of the canonically twisted groups, not a choice-free identification of \(\mu_n\) with \(\mathbf Z/n\).

Solution 2. Choose a finite triangulation of the compact analytic manifold and let \(C^\bullet\) be its integral simplicial cochain complex, as in Section 14. For every \(m\), Theorem 11.1 gives a natural, coefficient-compatible isomorphism \(H^q_{\mathrm{\acute et}}(X,\mathbf Z/\ell^m)\simeq H^q(C^\bullet/\ell^m)\). The finite cohomology towers satisfy Mittag–Leffler, so the Milnor \(\varprojlim^1\) term is zero. The degreewise surjective cellular tower has derived limit \(C^\bullet\otimes\mathbf Z_\ell\). This proves (14.1). Flat tensoring with \(\mathbf Q_\ell\) commutes with its cohomology; since the integral complex is finite free, the resulting group is \(H^q(C^\bullet\otimes\mathbf Q)\otimes_{\mathbf Q}\mathbf Q_\ell\). Its dimension is \(b_q\), proving the assertion. In contrast, the exact sequence

shows the two torsion contributions. Comparison equates the two finite-coefficient theories, but cannot remove those terms from their dimension.

Solution 3. Pullback along \(\epsilon\) is exact and preserves tensor products and their symmetry. The ordinary comparison is the adjoint of the pulled-back counit, so evaluation and cup diagrams commute. The compact comparison is built with the same adjunction after exact \(j_!\); its projection diagram commutes as well. Thus mixed compact/ordinary products compare, and exchange of degrees \(a,b\) has the common Koszul sign \((-1)^{ab}\).

For the top degree use \(\Lambda(d)\), not untwisted \(\Lambda\). Proper-support base change on both sides (Theorem 9.1) identifies the stalks of \(R^{2d}f_!\Lambda(d)\) and of its analytic counterpart at a complex point of \(S\) with the top compactly supported groups of the fibre, compatibly with comparison. By Theorem 10.1(3), on every fibre the stalk of the trace is integration over the complex orientation of the fibre when twists are trivialized by \(\zeta_n\), and is \((-1)^d\) times integration when they are trivialized by \(\zeta_n^{-1}\). A compatible root \(\zeta_n\) trivializes the Tate twist analytically; without it retain the twist.

Solution 4. For a smooth projective curve, degree zero follows by applying full faithfulness of Riemann existence to a trivial finite cover. Degree one follows by applying the equivalence to finite \(\Lambda\)-torsors with their group actions. Degree two follows from natural Kummer diagrams and Picard GAGA: algebraic \(H^2(\mathbf G_m)=0\); analytic \(H^2(\mathcal O^*)=0\) follows from the exponential sequence, Dolbeault vanishing above one and surface cohomology vanishing above two. Both Kummer cokernels are \(\operatorname{Pic}/n\). Higher constant groups vanish on both sides.

For every constructible sheaf on a proper curve, choose a dense normal open trivializing its finite monodromy. Embed its generic finite module into a finite sum of constant modules after a finite étale cover. Finite normalization extends the cover over the curve, and \(j'_*\Lambda=\Lambda\) on that normal cover. Add the boundary stalk sheaves to make the map injective on the whole curve. Repeat on the constructible cokernel. Finite pushforward is exact and comparison-compatible, so every term of this right resolution compares by the constant smooth projective curve theorem. In a fixed total degree the first-quadrant hypercohomology spectral sequence uses only finitely many terms and its comparison proves the assertion for the original sheaf.

Induct now on proper fibre dimension. Proper base change on both sides reduces a relative map to proper complex fibres. For a fibre of dimension \(d\geq2\), choose a rational function nonconstant on every positive-dimensional component and close its graph in \(X\times\mathbf P^1\). Its map \(a\) to \(\mathbf P^1\) has fibre dimension at most \(d-1\); the structural map \(b\) of \(\mathbf P^1\) has fibre dimension one. Induction compares \(Ra_*j'_!F_U\), proper constructibility permits comparison of its cohomology sheaves along \(b\), and the finite Leray filtrations compare its global cohomology on the graph. The proper graph projection is an isomorphism over \(U\); its pushforward of \(j'_!F_U\) is exactly \(j_!F_U\), because its other fibres carry zero coefficients. The boundary \(X\setminus U\) has smaller dimension, so localization finishes the comparison on \(X\). Proper base change then finishes the relative induction.

Express an arbitrary torsion sheaf as a filtered colimit of constructible finite torsion sheaves. Coherent étale continuity and compact Hausdorff continuity on the proper analytic fibres justify passage to that colimit on both sides. Finally compactify the given separated \(f\) and apply the proper result to \(j_!F\). This proves its arbitrary-torsion \(Rf_!\) comparison, independently of higher-dimensional resolution.

For ordinary constructible comparison, first use compact comparison, cup products, étale duality and classical duality on the complex manifold to compare constant coefficients on smooth schemes. Use resolution of singularities for a reduced \(X\). The cone of \(\Lambda_X\to Rp_*\Lambda_Y\) is bounded constructible and supported on the lower-dimensional complement of the open over which the proper resolution is an isomorphism. Proper comparison identifies its analytic cone; support induction and smooth constant comparison therefore prove constant comparison on \(X\). The finite-cover right resolution in arbitrary dimension then proves ordinary comparison for every constructible sheaf. Without constructibility this last argument does not apply, and the direct-sum/product example in Section 12 proves that its conclusion would be false.

17. Sources, proof dependencies and status

The historical comparison result is due to Artin; see SGA 4½, Arcata, IV, 6.3. Its concise reduction to proper smooth curves is expanded here into explicit coefficient resolutions, the proper graph induction, and filtered-colimit arguments. SGA 4, Exposé XVI, Section 4, including Theorem 4.1 and Lemmas 4.2–4.6, proves a more general relative comparison and uses resolution of singularities for the nonproper case. This lesson proves the absolute statement; the relative theorem is not proved here.

• [Riemann existence] The Riemann existence theorem, Theorems 2.1 and 3.2, proved in its course without normality hypotheses; the historical source is SGA 1, Exposé XII, Theorem 5.1.
• [Curve GAGA] Serre's comparison theorems and Chow's theorem, Theorems 4.1 and 5.2. The projective-curve Picard consequence is the input used in Section 5; no general coherent analytic comparison is substituted for the torsion comparison proved here.
• [Algebraic base change] Proper base change, Tag 095T and smooth base change, Tag 0EYU. The first is used in every proper fibre calculation. The second checks the smooth framework from Lesson 15; it alone cannot compare a nonproper analytic boundary stalk.
• [Classical sheaf tools] Proper base change in topology, cohomology chapter, including the compact-neighbourhood refinement argument preceding it. J. Anschütz, Lecture notes on étale cohomology, Theorem 4.16 and Lemma 4.20, PDF pages 25–27, give proper base change and compact Hausdorff continuity.
• [Alternative comparison exposition] J. S. Milne, Lectures on Étale Cohomology, version 2.21, Section 21, PDF pages 130–134. Its analytic effacement through elementary fibrations is a useful alternative; the present proper-support proof instead uses graph modifications, so no elementary-fibration lemma is silently assumed.
• [Resolution of singularities, Section 11] Resolution of singularities in characteristic zero, Corollary 5.1 (proper smooth models), proved in its course; Temkin, Functorial desingularization of quasi-excellent schemes in characteristic zero: the non-embedded case, arXiv:0904.1592, treats the general quasi-excellent case.
• [Classical topology] Poincaré duality on manifolds and Sheaf cohomology and singular cohomology on manifolds prove duality, dimension, integration and the sheaf-singular comparison used in Sections 5, 10 and 14, and Orientations and fundamental classes the orientations of complex manifolds, their products and local biholomorphisms used in Theorem 10.1; A. Hatcher, Algebraic Topology, Section 3.3, is a freely available account of manifold duality.

Within this course, Lesson 11 supplies the constructible and continuity machinery, Lesson 13 proper base change, and Lesson 14 compactification and proper support. Lesson 17 fixes the curve trace and coefficient self-injectivity. Lesson 18 supplies smooth duality, the point/divisor Gysin normalization and the projective-space computation. Their stated geometric and higher-trace inputs retain their stated status.

The Artin proper-support theorem and the ordinary constructible theorem are proved in Sections 5–11; neither is relegated to an unproved citation. Riemann existence, curve GAGA, resolution of singularities and the classical topological facts are proved in the lessons linked in Section 1. All examples and all four exercises have full solutions.