Traces of perfect complexes and Lefschetz numbers

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).

The cohomology groups of a flat finite-coefficient sheaf need not be projective over the coefficient ring. An ordinary matrix trace on those groups may therefore be undefined. The right object is the whole compact-support cohomology complex. We prove that it is perfect, define its trace, and establish the additivity and group-action identities needed to use that trace.

The coefficient ring may be noncommutative. This is essential when a finite covering introduces a group ring. The trace will take values in an additive quotient, and the group-action formulas will hold even when the group order is not invertible.

We use cohomological grading and left modules. The prerequisites are the course on derived categories and sheaf operations, compact-support cohomology and its finiteness and cohomological-dimension inputs, and the Frobenius conventions of the preceding lesson. Every algebraic trace argument below works over an arbitrary unital ring unless a hypothesis is stated.

1. The additive trace quotient

Definition 1.1. For a ring \(\Lambda\), let

where the denominator is the additive subgroup generated by \(ab-ba\), for \(a,b\in\Lambda\). The quotient is an abelian group. We do not regard it as a quotient ring: the subgroup need not be an ideal. If \(\Lambda\) is commutative, \(\Lambda^\natural=\Lambda\).

Write elements of a finite free left module as row vectors. A linear endomorphism acts by right multiplication by its matrix, and we define its trace as the class of the sum of the diagonal entries. This convention reverses the matrix order for composition; the following calculation is unchanged:

Here rectangular matrices are allowed. Thus maps \(f:\Lambda^m\to\Lambda^n\) and \(g:\Lambda^n\to\Lambda^m\) satisfy

For a finite projective \(P\), choose a split inclusion

and define

Proposition 1.2. Definition (1.3) is independent of the split inclusion. It is additive in \(u\), invariant under isomorphism, and satisfies

for finite projectives \(P,Q\) and maps \(f:P\to Q\), \(g:Q\to P\).

Proof. For a second splitting \(P\xrightarrow{j}\Lambda^n\xrightarrow{q}P\), apply (1.2) to \(iuq:\Lambda^n\to\Lambda^m\) and \(jp:\Lambda^m\to\Lambda^n\). Their two composites are \(iup\) and \(juq\), because \(qj=pi=1\). This proves independence. The same calculation after inserting splittings for both \(P\) and \(Q\) proves (1.4). Addition is immediate from diagonal sums. Conjugation by an isomorphism is a special case of (1.4). \(\square\)

For a direct sum, a block matrix has trace equal to the sum of the traces of its diagonal blocks. This remains true for finite projective summands by embedding them in free modules. In particular, if

is an exact sequence of finite projectives and \(u\) preserves it, choose a splitting. Its matrix is block triangular, so

The splitting need not commute with \(u\).

If a commutative ring \(A\) maps into the centre of \(\Lambda\), then \(\Lambda^\natural\) is an \(A\)-module. Indeed \(a[b,c]=[ab,c]\) for central \(a\), so multiplication by \(a\) preserves the commutator subgroup. This will make the products in the group-action formula meaningful.

For a commutative \(A\) and finite group \(G\),

To prove it, send a group-ring element to the sum of its coefficients on each conjugacy class. Products \(gh\) and \(hg\) are conjugate, so this map kills commutators. Conversely conjugate elements differ by a commutator:

After imposing these differences, the remaining quotient is free on the conjugacy classes, proving (1.6). This is why a noncommutative trace can record more than one ordinary scalar.

2. Traces in the perfect derived category

Definition 2.1. A complex in \(D(\Lambda)\) is perfect if it is isomorphic to a bounded complex \(P^\bullet\) of finite projective left \(\Lambda\)-modules. For a chain endomorphism \(u\) of such a complex, put

The sum is finite. A shift by one changes its sign.

Theorem 2.2. For \(K\in D_{\mathrm{perf}}(\Lambda)\) and \(u\in\operatorname{End}_{D(\Lambda)}(K)\), (2.1) defines an element of \(\Lambda^\natural\) independent of the chosen projective complex and chain representative. For maps \(f:K\to L\), \(g:L\to K\) between perfect complexes,

Proof. A bounded-above complex of projective modules is K-projective. Indeed, for a map \(f:P\to A\) to an acyclic complex, start in the highest degree \(b\). The map \(f^b\) lands in the cycles of \(A^b\); projectivity lifts it through \(A^{b-1}\twoheadrightarrow Z^b(A)\), giving \(h^b\). Having constructed the higher homotopy components, the map \(f^i-h^{i+1}d_P^i\) lands in \(Z^i(A)\), by the chain equation and the already established homotopy equation in degree \(i+1\). Lift again by projectivity. Descending induction gives \(f=d_Ah+hd_P\). Apply this argument also to every shift of \(A\); the Hom complex is acyclic. Consequently a quasi-isomorphism induces an isomorphism on homotopy classes out of \(P\). The roof argument in Flat modules and K-flat resolutions, Theorem 3.4 identifies these classes with derived morphisms. Thus every \(u\) has a chain representative, and an isomorphism between projective models has a homotopy inverse.

If a difference of representatives is \(dh+hd\), its trace is zero. In degree \(i\) it is

By (1.4), the trace of \(d^{i-1}h^i\) on \(P^i\) equals the trace of \(h^id^{i-1}\) on \(P^{i-1}\). The signs of these two appearances in (2.1) are opposite. Summing over the bounded range cancels every term.

Suppose \(Q^\bullet\) is another model. An isomorphism between these two bounded projective models is represented by a homotopy equivalence \(a:P^\bullet\to Q^\bullet\), with inverse \(b\). The transferred endomorphism is \(aub\). Degreewise cyclicity gives

Since \(ba\) is homotopic to the identity, homotopy invariance makes the right side \(\operatorname{Tr}(u;P^\bullet)\). This proves independence. Representing \(f,g\) on projective models and applying (1.4) in each degree proves (2.2). \(\square\)

Example 2.2a (a lift need only commute up to homotopy). Let \(P=k[0]\), let \(M=[k\xrightarrow{d}k^2]\) in degrees \(-1,0\), with \(d(t)=(0,t)\), and let the quasi-isomorphism \(\rho:P\to M\) send \(v\) to \((v,0)\). Set \(\alpha^{-1}=1\) and \(\alpha^0(x,y)=(x,x+y)\). This is a chain endomorphism. No endomorphism \(\beta\) of \(P\) satisfies \(\rho\beta=\alpha\rho\) as chain maps: the latter has second coordinate \(v\). But \(\beta=1\) gives equality up to the homotopy \(h:P^0\to M^{-1}\), \(h(v)=v\), since \(\alpha\rho-\rho=d h\). Thus the commuting diagram in Milne's Proposition 29.17 must be interpreted in the homotopy or derived category. Theorem 2.2 supplies precisely that statement and its trace independence.

If a perfect complex has finite projective cohomology modules, its trace is the alternating sum of their projective traces. To prove this, use a bounded projective model and descend from its top degree. Splitting the quotient onto the projective top cohomology makes the top boundary projective. Splitting the preceding map onto that boundary makes the preceding cycles projective, and splitting off the projective cohomology makes the next boundary projective. Continue downwards. The complex is then a direct sum of its cohomology and contractible two-term summands; homotopy invariance removes the latter. In particular, over a field \(E\),

Over a general ring the cohomology modules may not be projective, so (2.3) is not a definition. The projective-complex trace (2.1) is the definition. If \(R\) is a commutative integral domain with fraction field \(E\), extension of a finite projective model and flatness of \(E/R\) show that its trace maps to \(\sum_i(-1)^i\operatorname{Tr}_E(H^i(u)\otimes_R E;H^i(K)\otimes_R E)\). Since \(R\to E\) is injective, that scalar determines the original trace. No projectivity of the cohomology over \(R\) is needed for this fraction-field computation.

What cohomology can fail to see

Lemma 2.3 (a finite bound for cohomologically zero maps). Let \(S=\{s_1<\cdots<s_r\}\) be a finite set of degrees. Suppose the complexes in

have cohomology only in \(S\), and \(H^q(f_j)=0\) for every \(q,j\). Then \(f_r\cdots f_1=0\) in the derived category. In particular, a cohomologically zero endomorphism of a complex supported in \([a,b]\) has \(u^{b-a+1}=0\).

Proof. The truncation orthogonality used here can be checked by the projective-model argument: for a source with cohomology in degrees \(\leq q-1\), the downward cycle construction with arbitrary free sums gives a free model vanishing above \(q-1\). A target with cohomology in degrees \(\geq q\) has its good lower truncation vanishing below \(q\). Every chain map between these models is zero, and the bounded-above K-projectivity proof in Theorem 2.2 computes the derived Hom. For such a map \(f:E\to F\), the composite \(\tau_{\leq q}E\to\tau_{\leq q}F\to H^q(F)[-q]\) vanishes. Its restriction to \(\tau_{\leq q-1}E\) is zero by the orthogonality of the standard truncations. The truncation triangle therefore factors it through \(H^q(E)[-q]\), where the induced map is \(H^q(f)=0\). Exact Hom for the target truncation triangle now factors \(\tau_{\leq q}f\) through \(\tau_{\leq q-1}F\). Start with \(E_0\simeq\tau_{\leq s_r}E_0\). The first map factors through \(\tau_{\leq s_r-1}E_1\), which is canonically isomorphic to \(\tau_{\leq s_{r-1}}E_1\), since the intervening degrees vanish. Naturality of truncation allows the next map to lower the bound again. After \(r\) maps the composite factors through \(\tau_{\leq s_1-1}E_r=0\). If \(S\) is empty all objects are already zero. The interval bound follows by taking \(S=\{a,\ldots,b\}\). \(\square\)

If the ring \(R\) is commutative and \(K\) is perfect, a cohomologically zero endomorphism can still have nonzero trace. Its trace belongs to the nilradical of \(R\): Lemma 2.3 makes the endomorphism nilpotent; after derived extension to the residue field at any prime, it is still nilpotent. Its induced operators on the finite-dimensional cohomology are nilpotent, hence have trace zero. Equation (2.3), and extension of the finite projective trace model, show that \(\operatorname{Tr}_R(u;K)\) maps to zero in every residue field. The intersection of all prime ideals is the nilradical. Thus that trace is zero when \(R\) is reduced. The example in §3 shows why the reduced hypothesis matters.

3. Filtered additivity and its boundary

A finite decreasing filtration of a complex has subcomplexes \(F^pK\) and associated graded complexes

In the filtered derived category, we invert maps that are quasi-isomorphisms on every graded piece. Because the filtration is finite, they are also quasi-isomorphisms on the underlying complex. An endomorphism in this category includes compatible filtered data; forgetting the filtration loses some of that data.

Call a filtered complex filtered perfect if it has a bounded model whose terms have finite filtrations with finite projective graded modules. Such a module is itself finite projective, because its filtration splits successively. Equivalently, all the graded complexes are perfect and only finitely many are nonzero.

Here are the model facts behind this equivalence and the trace argument. Starting with the last filtration step, replace its graded complex by a bounded projective model. To add the next graded piece, its extension is specified by a morphism from that graded piece to the already constructed filtered tail shifted by one. A projective model for the graded piece represents this morphism by a chain map. Its cone, shifted back by one, has terms that are direct sums of the old model and the new projective terms. Giving it the old tail filtration and the new quotient constructs the desired model. Induction through the finite filtration proves existence.

Moreover maps from this model in the filtered derived category are represented by filtered chain maps modulo filtered homotopy. To check this, let \(C\) be filtered acyclic, meaning that every \(\operatorname{gr}^pC\) is acyclic. Its finite filtration makes every \(F^pC\) acyclic. For a filtered projective term, a splitting expresses the complex of filtered maps into \(C\) as a finite sum of complexes \(\operatorname{Hom}_\Lambda(Q,F^pC)\), with \(Q\) projective. These are acyclic. The source complex is bounded, so the whole Hom complex is acyclic. Thus filtered quasi-isomorphisms do not change the morphisms out of this model, proving the assertion.

Theorem 3.1 (filtered additivity). If \((K,F)\) is filtered perfect and \(u\) is its endomorphism in the filtered derived category, then

Proof. Use the bounded filtered projective model just described, and a filtered chain representative for \(u\). In each degree split the finite filtration. The matrix of \(u^i\) is block triangular and its diagonal blocks are the maps on the graded modules. Equation (1.5) gives their trace sum. Sum this identity with the sign \((-1)^i\), and interchange the two finite sums. The result is (3.1). Theorem 2.2 makes it independent of the model. \(\square\)

This also applies after a derived sheaf functor. Module sheaves on the small étale site form a Grothendieck abelian category: filtered colimits are exact on geometric stalks, and the sum of the \(a_!\Lambda\) over a set of affine étale basis objects is a generator. Thus K-injective resolutions in Grothendieck categories, Theorem 2.4 supplies enough injectives. The bounded-below resolution construction is Injective modules and bounded-below derived functors, Theorem 4.1(1). Resolve a finite filtered sheaf complex by a bounded-below filtered injective complex. Each term has injective graded pieces, hence a split filtration. A left exact additive functor preserves these finite split sequences, so applying it and then taking a graded piece gives the same complex as applying it to that graded piece. Consequently

Filtered injective resolutions can be built one filtration step at a time from ordinary injective resolutions; adjoining a finite direct sum at each step gives the split inclusions needed here. This is the filtered-resolution version of the ordinary derived construction. For compact support, apply it to \(R\Gamma(\overline X,-)\) after the exact filtered functor \(j_!\). Hence a short exact sequence of sheaf complexes with its natural two-step filtration gives the compatible filtered compact-support complexes. When the graded pieces are perfect, (3.1) proves their trace additivity.

Why an arbitrary triangle needs care

Over a ring with nilpotents, a morphism of distinguished triangles need not have the trace relation of a compatible cone or filtered endomorphism. Let

where \(C=[R\xrightarrow{\epsilon}R]\) has terms in degrees \(-1,0\). Give the first two objects the zero endomorphism. On \(C\), define

This is a chain map because \(\epsilon^2=0\). One has \(u_Ci=0\). The map \(pu_C:C\to R[1]\) is null-homotopic: the homotopy \(C^0\to R[1]^{-1}\) given by the identity has \(hd=\epsilon\) in degree \(-1\). Thus the three maps form a morphism of the distinguished triangle in \(D_{\mathrm{perf}}(R)\). Their traces are

Since \(R\) is commutative and \(\epsilon\ne0\), the required relation \(\operatorname{Tr}(u_C)=\operatorname{Tr}(u_R)-\operatorname{Tr}(u_R)\) fails. This endomorphism does not lift to the corresponding filtered triangle.

Over a field, however, no such example exists. A morphism of distinguished triangles gives an endomorphism of its finite long exact cohomology sequence. The alternating sum of traces on a finite exact sequence of vector spaces is zero: split each term into the image of its incoming map and a complement mapping isomorphically to the next image; their trace contributions cancel. Grouping the terms of the long exact sequence by the three complexes and using (2.3) gives

Thus the failure in (3.4) concerns general coefficient rings, precisely the situation in which we need the filtered formalism.

Proposition 3.2 (arbitrary triangles over a reduced ring). For any commutative ring \(R\), the trace defect of a morphism of distinguished triangles of perfect complexes,

belongs to the nilradical. Consequently \(\Delta=0\) over a reduced ring.

Proof. For each prime \(\mathfrak p\), derived tensor with \(\kappa(\mathfrak p)\) takes the given triangle and its morphism to such a triangle over a field. It takes a bounded finite projective trace model to its scalar extension, so its trace is the image of the original trace. The field proof of (3.5) gives \(\Delta=0\) in every \(\kappa(\mathfrak p)\). Hence \(\Delta\) lies in every prime ideal, proving the claim. \(\square\)

In (3.3), the cone map also has zero action on both cohomology modules: in degree \(-1\) those cycles are \(\epsilon R\), killed by multiplication by \(\epsilon\), and in degree zero its component is zero. It squares to zero as a chain map, yet its trace is \(-\epsilon\). A filtered or compatible cone endomorphism retains the data that gives exact additivity over arbitrary rings; cohomological vanishing alone supplies only the nilpotence conclusion above in the commutative case.

4. Perfectness of compact-support cohomology

Definition 4.1. A complex of left \(\Lambda\)-modules has Tor amplitude in \([a,b]\) if

for every right \(\Lambda\)-module \(N\). The sheaf version tests right-module sheaves, or equivalently geometric stalks with a uniform bound. For a left Noetherian coefficient ring, the category \(D_{\mathrm{ctf}}(X,\Lambda)\) consists of complexes with constructible cohomology and locally finite Tor amplitude. On a quasi-compact \(X\), finitely many local bounds give a uniform interval.

Lemma 4.1a (forgetting the coefficient structure). Étale cohomology of a sheaf of left modules over a constant ring agrees, as an abelian group, with cohomology of its underlying abelian sheaf. The same holds for compact support. Commutativity of the ring is unnecessary.

Proof. Choose a set of geometric points detecting sheaf isomorphisms and use the resolution that begins with \(\mathcal F\to\prod_x x_*\mathcal F_x\), then repeats this construction on the cokernels. The initial map is injective on stalks, and all these operations commute with forgetting the module structure. Each product of point direct images is acyclic for global sections in both categories. To verify this, resolve each point module by injective modules. A point direct image preserves injectives, since its left adjoint, the stalk functor, is exact. Products of injectives are injective. This product resolution is exact on every étale object: point direct images there are products indexed by the lifts of the point, and products of exact sequences of modules are exact. Its global sections are the corresponding product of the point resolutions and have no positive cohomology. The same proof using injective abelian groups proves abelian-sheaf acyclicity. Thus the identical global-section complex of the displayed resolution computes cohomology in both categories. For compact support, extend by zero into a compactification, which commutes with forgetting, and apply this argument to its global sections. \(\square\)

The geometric construction of compact support uses additive, coefficient-linear functors. Its open/proper comparison and composition maps are defined by adjunction and restrictions of supports; the same construction therefore applies to left modules over a noncommutative ring. The tensor comparison for those rings is proved explicitly in (4.3).

Lemma 4.2 (algebraic criterion). Let \(\Lambda\) be left Noetherian. A complex is perfect if and only if it has finite Tor amplitude and every cohomology module is finitely generated.

Proof. A bounded finite projective complex plainly has finite Tor amplitude. Its kernels, images and cohomology are finitely generated by the Noetherian hypothesis.

Conversely choose a bounded-above complex \(P^\bullet\) of finite free modules representing \(K\). It can be constructed from the highest nonzero cohomology degree downwards: choose finitely many cycles generating that cohomology, map a finite free module to them, and kill the remaining cohomology of the cone at the next step. The new kernels are finitely generated because \(\Lambda\) is left Noetherian, so every step uses finite free modules. This gives a quasi-isomorphism after continuing indefinitely to the left.

Let \(a\) be a lower Tor bound, and put \(C=P^a/\operatorname{im}(d^{a-1})\). The left tail of \(P^\bullet\) is a free resolution of \(C\), since \(K\) has no cohomology below \(a\). For every right module \(N\),

Thus \(C\) is flat. It is finitely presented: the image of \(d^{a-1}\) is finitely generated inside the finite free \(P^a\).

Here is the finitely presented flat-to-projective argument, including the noncommutative case. In a presentation \(0\to L\to\Lambda^m\to C\to0\), let \(r_a=\sum_j a_{aj}e_j\) be finitely many generators of \(L\). Flatness of \(C\) makes \(N\otimes_\Lambda L\to N\otimes_\Lambda\Lambda^m\) injective for every right module \(N\). Take \(N\) to be the cokernel of the map of finite free right modules sending the \(j\)-th basis element to \(\sum_a e_a a_{aj}\). The vector \((r_a)_a\in L^{\oplus s}\) maps to zero in \(N\otimes_\Lambda\Lambda^m\), since the equations \(\sum_j a_{aj}y_j=r_a\) have the solution \(y_j=e_j\) there. Injectivity makes it zero also in \(N\otimes_\Lambda L\). Right exactness of tensor then gives a solution with every \(y_j\in L\). The assignments \(\overline e_j\mapsto e_j-y_j\) satisfy all the generating relations and define a section \(C\to\Lambda^m\). This proves projectivity.

Replacing the left tail by \(C\) in degree \(a\) now gives a bounded complex of finite projectives quasi-isomorphic to \(K\). \(\square\)

Lemma 4.2a (arbitrary-ring criterion). For an arbitrary unital ring, call a complex pseudocoherent if it has a bounded-above model of finite projective modules. A complex is perfect if and only if it is pseudocoherent and has finite Tor amplitude. If its Tor amplitude lies in \([a,b]\), its finite projective model can be chosen to have terms only in that interval.

Proof. The forward implication follows from the finite projective model. For the reverse, take the pseudocoherent model \(P\). Its cohomology is zero above \(b\). If its top nonzero term is in degree \(c>b\), the differential onto that projective term is surjective and splits. Remove the resulting contractible two-term direct summand. Its complement still has finite projective terms. Repeat finitely many times to arrange \(P^i=0\) for \(i>b\). At the lower Tor bound \(a\), the same tail computation as in Lemma 4.2 makes \(C=P^a/\operatorname{im}(d^{a-1})\) flat. It is finitely presented: \(P^a\) is finitely presented and the image of the finitely generated \(P^{a-1}\) is generated by finitely many relations. The flat-to-projective proof above applies without Noetherianity. Truncate to \(C\to P^{a+1}\to\cdots\to P^b\), the required finite projective model. \(\square\)

Finite generation of cohomology cannot replace pseudocoherence over an arbitrary ring. Even the necessity would fail: let \(R=k\oplus V\), where \(V^2=0\) and \(V\) has infinite dimension over \(k\). For \(0\ne v\in V\), the perfect complex \([R\xrightarrow{v}R]\) in degrees \(-1,0\) has \(H^{-1}=V\). Any finite list of elements generates only its finite-dimensional \(k\)-span, so this cohomology module is not finitely generated. Lemma 4.2 avoids this issue by imposing left Noetherianity.

Lemma 4.2b (finite flat generators and lifting). On a Noetherian scheme \(X\), for a left Noetherian ring \(\Lambda\), a constructible module sheaf is finitely presented by finite sums of \(P_U=a_!\underline\Lambda_U\), with \(U\) affine and \(a:U\to X\) étale of finite presentation. Every epimorphism \(\mathcal E\twoheadrightarrow\mathcal B\) onto a constructible sheaf admits a map from such a finite sum to \(\mathcal E\) whose composite is onto. In particular, bounded-above constructible complexes have bounded-above resolutions by these flat generators, with a finite sum in each degree.

Proof. The scheme argument in Constructible sheaves and extension by zero, Lemma 4.1 stratifies \(a\) into finite étale maps. Thus \(P_U\) is constructible, and its stalks are finite free left modules, so it is flat. This argument is independent of commutativity of the coefficient ring. On common trivializing étale covers, maps between finitely generated constant modules are constant after refining around their finitely many generators. Kernels are finitely generated by left Noetherianity; cokernels are finitely generated as quotients. Finite stratifications therefore give closure under kernels, cokernels and extensions, using a finite presentation for the quotient in an extension. These are exactly the module arguments of that provider's Lemma 3.1 and Proposition 3.2, with all coefficients acting on the left.

Every section over an affine étale object defines a map \(P_U\to\mathcal E\), by adjunction. The sum over all these sections surjects onto \(\mathcal E\), hence onto \(\mathcal B\). For finite partial sums, the cokernels of their maps to \(\mathcal B\) are constructible. Their supports are closed in the constructible topology of \(X\), decrease, and have empty intersection, since each stalk of \(\mathcal B\) is finitely generated. Compactness in that topology, proved in the provider's Lemma 5.2, makes some support empty. This gives the finite lifting assertion. Applying it to \(\mathcal B\), and then to the constructible kernel of the resulting finite surjection, gives a finite presentation. To resolve a bounded-above complex, successively lift finite generators of the cycle quotient in the next mapping cone. The lifting assertion supplies the differential and the map to the original complex together; it does not assume the flat sheaves are projective objects. The usual downward cone construction kills cohomology in one degree at each step and gives the claimed quasi-isomorphism. \(\square\)

At a lower Tor bound \(a\), truncate this flat resolution by \(P^a/\operatorname{im}(d^{a-1})\). Stalkwise the tail computes Tor, so this quotient is flat; it is constructible by the closure just proved. Thus \(K\in D_{\mathrm{ctf}}(X,\Lambda)\) has a bounded constructible flat model. Conversely such a model has constructible cohomology and finite Tor amplitude. Each geometric stalk is perfect by Lemma 4.2. This proves the coefficient-category characterization, including noncommutative left Noetherian rings, rather than relying on the omitted argument in the source.

Theorem 4.3 (all dimensions). Let \(X\) be separated and of finite type over an algebraically closed field of characteristic \(p\). Let \(\Lambda\) be a finite, possibly noncommutative ring whose cardinality is prime to \(p\), and \(K\in D_{\mathrm{ctf}}(X,\Lambda)\). Then

If \(K\) has Tor amplitude in \([a,b]\) and \(d=\dim X\), the compact-support complex has Tor amplitude in \([a,b+2d]\).

Proof. Write \(L=R\Gamma_c(X,K)\). The compact-support finiteness prerequisite says that the cohomology of a constructible finite torsion sheaf is finite, and its compact-support cohomology vanishes above \(2d\). The bound holds for every torsion sheaf killed by an integer prime to \(p\), without constructibility. The hypercohomology spectral sequence

therefore has finitely many nonzero rows and columns, all finite. Hence \(L\) has bounded finite cohomology.

We need the coefficient projection formula for every right \(\Lambda\)-module \(N\):

Here \(N\) is used as a constant right-module sheaf on the right. We give the resolution argument, including the boundedness needed for arbitrary \(N\).

Choose a compactification \(j:X\hookrightarrow\overline X\) with \(X\) dense, so that \(\dim\overline X=d\). Put \(J=j_!K\). Extension by zero is exact, preserves flatness, and satisfies \(j_!(N\otimes_\Lambda^LK)=N\otimes_\Lambda^LJ\), as can be checked on stalks. Thus (4.3) reduces to the ordinary global-section projection formula on the proper \(\overline X\).

On \(\overline X\), torsion cohomology commutes with filtered colimits by étale continuity, hence with arbitrary direct sums, and has cohomological dimension \(D=2d\). Choose a bounded complex \(I^\bullet\) of \(\Gamma(\overline X,-)\)-acyclic sheaves representing the bounded \(J\). Resolve bounded below by injectives and, above the last cohomology degree, truncate after \(D\) further steps at the cycle sheaf. Dimension shifting shows that this last cycle sheaf is also acyclic. All terms remain annihilated by an integer prime to \(p\).

Choose a free right-module resolution \(Q^\bullet\to N\) in nonpositive degrees. Each \(Q^i\otimes_\Lambda I^j\) is a direct sum of copies of \(I^j\), so it is acyclic for global sections on \(\overline X\), and its sections are \(Q^i\otimes_\Lambda\Gamma(\overline X,I^j)\). Since \(I^\bullet\) is bounded, every total degree has only finitely many contributing \(j\). The bounded-above total complex computes \(N\otimes_\Lambda^LJ\). Applying \(R\Gamma(\overline X,-)\) to this total complex is computed termwise by global sections: truncate farther and farther to the left; finite cohomological dimension makes these truncations stabilize in each degree. Thus

This is (4.3), because a bounded-above complex of free right modules computes the derived tensor product.

By the Tor bound on \(K\), the sheaf complex \(N\otimes_\Lambda^LK\) has cohomology only in \([a,b]\). The same cohomological-dimension spectral sequence, now without any finiteness assumption on \(N\), puts its compact-support cohomology in \([a,b+D]\). Equation (4.3) proves this as a uniform Tor interval for \(L\). Finally a finite ring is left Noetherian, and its finite cohomology modules are finitely generated. Lemma 4.2 proves (4.1). \(\square\)

This proves the perfectness statement in every dimension, with noncommutative finite coefficients. [Stacks, Tag 03TV] discusses the projective-curve case; Deligne's [Rapport], §4.9, gives the direct-image formalism underlying the argument. The use of the full uniform cohomological-dimension bound in (4.3) is what controls the unbounded free resolution of \(N\).

Noetherian torsion coefficients

Theorem 4.4. Let \(X\) and \(d\) be as in Theorem 4.3. Let \(\Lambda\) be any left Noetherian unital ring killed by an integer \(n\) invertible in the base field. For \(K\in D_{\mathrm{ctf}}(X,\Lambda)\), the complex \(R\Gamma_c(X,K)\) is perfect. Its Tor amplitude is contained in \([a,b+2d]\) if that of \(K\) is contained in \([a,b]\). Formula (4.3) holds for every right \(\Lambda\)-module. Finite cardinality of \(\Lambda\) is unnecessary.

Proof. The bounded acyclic model, arbitrary free resolution and finite-dimension argument proving (4.3) use only that all sheaves are killed by \(n\). They therefore apply unchanged. They give the claimed Tor bound. We supply the finite-generation argument for these larger coefficients.

Put \(A=\mathbf Z/n\). For each separated affine étale \(U\to X\) in Lemma 4.2b, Theorem 4.3 makes \(C_U=R\Gamma_c(U,A)\) perfect over \(A\). The central map \(A\to\Lambda\) and that theorem's coefficient projection formula give

A bounded finite projective \(A\)-model for \(C_U\) becomes a bounded finite projective left \(\Lambda\)-model after this tensor product: tensor the finite free splitting of each term. Compact-support composition and exact étale extension by zero identify (4.4) with \(R\Gamma_c(X,P_U)\). Consequently every \(H^s_c(X,P_U)\) is finitely generated over \(\Lambda\).

Resolve \(K\) by the bounded-above finite sums \(P^q\) from Lemma 4.2b. The compact-support dimension bound \(D=2d\), valid for all \(n\)-torsion sheaves, makes the term spectral sequence

converge with a finite filtration in each total degree. Here is the boundedness justification: cut the source below degree \(m\) by the brutal truncation, retaining its original terms in degrees \(\geq m\). The omitted quotient lies in degrees \(<m\), so its compact-support cohomology lies in degrees \(\leq m-1+D\). For any fixed output degree, sufficiently negative \(m\) makes both adjacent terms in the quotient's long exact sequence zero. Thus the finite spectral sequences stabilize to (4.5). Only \(q=i-D,\ldots,i\) occur at total degree \(i\), and differentials beyond length \(D+1\) vanish because they leave the strip. Every entry is finitely generated; subquotients and finite extensions remain so by left Noetherianity. Hence \(H^i_c(X,K)\) is finitely generated. Its degrees are bounded by the Tor bound already established. Lemma 4.2 proves perfectness. \(\square\)

For instance \(\mathbf F_\ell[t,t^{-1}]\), with \(\ell\ne p\), satisfies this theorem. Constructible modules over it mean finitely generated modules on finitely many locally constant pieces. Their underlying sets can be infinite; no finite-monodromy claim was used in the proof. Deligne's Rapport, §4.9, gives the historical direct-image perfectness theorem. Theorem 4.4 supplies its absolute compact-support argument here and retains the explicit cohomological bound.

5. Global and local Lefschetz numbers

Let \(X_0\) be separated and finite type over \(\mathbf F_q\), \(X=X_0\times\overline{\mathbf F}_q\), and \(K_0\in D_{\mathrm{ctf}}(X_0,\Lambda)\), with \(\Lambda\) as in Theorem 4.4. The natural sheaf correspondence from the preceding lesson extends termwise, and hence in the derived category, to

It gives an endomorphism of the perfect complex \(R\Gamma_c(X,K)\). Define

Each geometric stalk \(K_{\bar x}\) is perfect by §4. For a rational point \(x\), local geometric Frobenius from the preceding lesson acts on it. Define

The sum is finite by the fixed-point theorem. Independence of the geometric point follows from conjugacy and Theorem 2.2. These definitions make sense before proving the equality of (5.1) and (5.2). That equality is the trace formula to be established in later lessons.

Both definitions respect a finite filtered endomorphism with \(D_{\mathrm{ctf}}\) graded pieces. For the global number, (3.2), Theorem 4.4 and filtered additivity apply to \(R\Gamma_c\). For the local number, stalks are exact and preserve the filtration, so (3.1) applies to each stalk. Both numbers also commute with extension to another left Noetherian torsion coefficient ring as in Theorem 4.4: (4.3) identifies the extended global complex, and extension of a finite projective model keeps it finite projective. Its diagonal trace is the image of the original trace under the induced map of additive quotients.

For \(\Lambda=\mathbf Z/\ell^s\), the constant sheaf on \(\mathbf P^1\) has \(H^0=\Lambda\), \(H^2=\Lambda(-1)\), and no other cohomology. These are the projective-space cohomology results used in Lesson 1. Geometric Frobenius acts by \(1\) and \(q\), respectively, because it acts by \(q^{-1}\) on the positive Tate twist. Thus

Every rational stalk is the trivial module of rank one, so the local number is \(q+1\) as well. Both numbers are elements of \(\Lambda\); the integer point count is reduced modulo \(\ell^s\). This example checks the convention, and does not prove the general trace formula.

6. Group actions and the integral trace formulas

Let \(A\) be commutative and map into the centre of a possibly noncommutative \(\Lambda\). Let \(G\) be finite. A monoid extension of \(G\) by \(\mathbf N\) means the nonnegative part \(\Gamma\) of a group extension

Choose a lift \(F\in\Gamma\) of \(1\). Conjugation by \(F\) in \(\widetilde\Gamma\) defines an automorphism \(\alpha\) of \(G\), and \(Fg=\alpha(g)F\). A \(\Gamma\)-module gives an action of \(G\) and a compatible operator \(F\); this operator need not be invertible. Every element above \(1\) is \(gF\), for a unique \(g\in G\).

For any subgroup \(H\subset G\), the coefficient of the identity defines

It is well defined: the identity coefficient of a commutator of \(\sum a_hh\) and \(\sum b_hh\) is

a sum of commutators in \(\Lambda\). If \(Q\) is finite projective over \(\Lambda[H]\) and \(u\) is \(H\)-linear, define its \(H\)-trace by

Lemma 6.1 (restriction and tensor identities).

1. An \(H\)-linear endomorphism of a finite projective \(\Lambda[H]\)-module satisfies

   \[ \operatorname{Tr}_\Lambda(u;Q) =|H|\operatorname{Tr}_\Lambda^H(u;Q). \tag{6.1} \]

2. If \(P\) is finite projective over \(A[G]\) and \(M\) is a \(\Lambda[G]\)-module finite projective over \(\Lambda\), then \(P\otimes_A M\), with diagonal \(G\)-action, is finite projective over \(\Lambda[G]\).

3. If \(u\) on \(P\) and \(v\) on \(M\) commute with \(G\), then

   \[ \operatorname{Tr}_\Lambda^G(u\otimes v;P\otimes_A M) =\operatorname{Tr}_A^G(u;P)\, \operatorname{Tr}_\Lambda(v;M). \tag{6.2} \]

Proof. For (1), first use \(Q=\Lambda[H]\) and right multiplication by \(a=\sum a_hh\). In its free \(\Lambda\)-basis indexed by \(H\), every diagonal coefficient is \(a_1\), giving \(|H|a_1\). Finite sums give the free case. A projective summand has a split inclusion and projection; extending \(u\) by zero and using Proposition 1.2 gives the projective case.

For (2), begin with \(P=A[G]\). The untwisting isomorphism is

The target has its \(G\)-action only on the first factor. The inverse sends \(h\otimes m\) to \(h\otimes hm\), and these formulas check \(\Lambda[G]\)-linearity for the diagonal action on the source. The target is finite projective because \(M\) is a summand of a finite free \(\Lambda\)-module. Taking finite sums and the splitting of \(P\) proves (2).

For (3), extend \(u\) by zero from a projective summand of a finite free \(A[G]\)-module. Both sides respect the split inclusion and projection, so it suffices to compute a diagonal block when \(P=A[G]\). Write \(u=R_a\), right multiplication by \(a=\sum a_hh\). After (6.3), the operator is

For \(h\ne1\), its group-ring diagonal coefficient is supported at \(h\), so its \(G\)-trace is zero. For \(h=1\), it is \(a_1\operatorname{Tr}_\Lambda(v;M)\). To compute this also for projective \(M\), choose a finite free splitting of \(M\) after untwisting; the group acts only on the first factor, so the splitting gives a group-linear free model for the trace. The surviving \(a_1\) is \(\operatorname{Tr}_A^G(u;P)\). This proves (6.2). \(\square\)

Lemma 6.1a (the norm on projective group modules). For a finite projective \(\Lambda[G]\)-module \(Q\), the norm is an isomorphism

If \(Q\) also has the compatible forward \(F\)-action, this isomorphism intertwines it, and

Proof. For \(Q=\Lambda[G]\), coinvariants are one copy of \(\Lambda\); an invariant vector has equal coefficients on every group element, and norm sends its coefficient to precisely that invariant vector. This proves the isomorphism for finite free group modules. Invariants, coinvariants and norm preserve a split inclusion and retraction, so it follows for \(Q\). In particular both sides are finite projective over \(\Lambda\). Since \(Fh=\alpha(h)F\) and \(\alpha\) permutes \(G\), norm commutes with the induced forward action. Write \(q:Q\to Q_G\) for the quotient and \(a:Q_G\to Q\) for norm followed by inclusion. Then \(qa=|G|\), while \(\sum_{\gamma\mapsto1}\gamma=a\overline Fq\). Rectangular projective cyclicity gives its trace on \(Q\) as \(\operatorname{Tr}(qa\overline F;Q_G)=|G|\operatorname{Tr}(\overline F;Q_G)\). On invariants and coinvariants the sum of the lifts is \(|G|F\), and norm identifies their actions. This proves every equality without cancelling \(|G|\). \(\square\)

For \(\gamma\in\Gamma\), put

Restriction of a projective \(A[G]\)-module to \(A[Z_\gamma]\) is projective, since \(A[G]\) is finite free over that subgroup ring. The operator \(\gamma\) commutes with \(Z_\gamma\), so Lemma 6.1 gives

and, for \(P,M\) as above with \(\Gamma\)-actions,

These identities use no division by the group order.

Theorem 6.2 (coinvariant trace). Let \(Q\) be a \(\Lambda[\Gamma]\)-module finite projective over \(\Lambda[G]\). Then \(Q_G\) is finite projective over \(\Lambda\), the quotient monoid acts on it, and

The sum contains one representative of each \(G\)-conjugacy class in the fibre above \(1\). It holds over every \(\Lambda\), even if \(|G|\) is zero or a zero divisor.

Proof. Coinvariants are extension of scalars along the augmentation \(\Lambda[G]\to\Lambda\). A finite projective module stays finite projective under this extension. Normality of \(G\) makes the \(F\)-action descend; all lifts of \(1\) have the same quotient action.

We can reduce the formula to \(Q=\Lambda[G]^m\) without requiring a \(\Gamma\)-stable free complement with invertible \(F\). Choose \(G\)-linear splittings \(Q\xrightarrow{i}V\xrightarrow{p}Q\), with \(V=\Lambda[G]^m\). Extend the operator by \(F_V=iF_Qp\). It obeys \(F_Vh=\alpha(h)F_V\), so it defines the needed forward \(\Gamma\)-action. Both \(i,p\) intertwine the forward action. Its complement has zero \(F\)-action. Applying Proposition 1.2 to coinvariants and to every restricted centralizer trace shows that the two sides for \(V\) are the two sides for \(Q\).

Let \(e_1,\ldots,e_m\) be the free group-ring basis of \(V\), and write

The quotient basis has ordinary diagonal trace

Fix \(\gamma=gF\) and \(Z=Z_\gamma\). As a left \(\Lambda[Z]\)-module, \(V\) has basis \(he_i\) for representatives of \(Z\backslash G\). The operator sends

Its identity coefficient in the diagonal block for \(he_i\) occurs exactly when

Therefore

The map from these cosets to the \(G\)-conjugacy class of \(\gamma\) is

It is well defined and bijective: two \(h\)'s give the same conjugate precisely when their quotient on the left centralizes \(\gamma\). As the conjugacy classes above \(1\) vary, (6.10) partitions all the elements \(t^{-1}F\), one for each \(t\in G\). Hence summing (6.9) over those classes gives exactly (6.8). This proves (6.6) directly, without cancelling \(|G|\). \(\square\)

Changing a representative conjugates its centralizer and its module operator. The trace is transported through the corresponding group-ring isomorphism, whose identity-coefficient map is unchanged. Thus every summand in (6.6) is independent of the representative.

Corollary 6.3 (trace sorites). Let \(P\) be an \(A[\Gamma]\)-module finite projective over \(A[G]\), and \(M\) a \(\Lambda[\Gamma]\)-module finite projective over \(\Lambda\). Then \((P\otimes_A M)_G\) is finite projective over \(\Lambda\), and

Proof. Lemma 6.1 makes the diagonal tensor product projective over \(\Lambda[G]\). Apply Theorem 6.2 to it and use (6.5) term by term. The scalar from \(A\) acts on \(\Lambda^\natural\) through the central map \(A\to\Lambda\). \(\square\)

This proves the identities of [Stacks, Tags 03U5–03UD], including the integral coinvariant identity for which an averaged formula would be insufficient. In (6.11) the same element \(\gamma=gF\) acts on both \(P\) and \(M\). Replacing it by \(g\) on \(M\) would lose its Frobenius action.

The statements extend to bounded compatible complexes. Apply each identity to every pair of terms; the tensor term \(P^i\otimes_A M^j\) has sign \((-1)^{i+j}\), so the double finite sum gives the product of the two alternating traces. Homotopy invariance from Theorem 2.2 makes this independent of chain representatives. Filtered refinements retain the additivity of §3.

7. Exercises with complete solutions

Exercise 7.1 (easy: projective trace). Prove independence of a free splitting for the trace of a finite projective module, and cyclicity for maps between two such modules.

Solution. For splittings \(i,p\) and \(j,q\) of \(P\), the maps \(iuq\) and \(jp\) have composites \(iup\) and \(juq\), respectively. Their traces agree by the rectangular matrix calculation (1.1). For maps \(f:P\to Q\), \(g:Q\to P\), insert splittings for both modules. The resulting rectangular free-module maps have products extending \(gf\) and \(fg\) by zero, so their equal traces are precisely the two projective traces. This also proves \(\operatorname{Tr}(uv)=\operatorname{Tr}(vu)\) on a single module.

Exercise 7.2 (medium: triangles over a field and over a ring). Prove that no trace-additivity counterexample exists for a morphism of distinguished triangles in \(D_{\mathrm{perf}}(k)\), where \(k\) is a field. Then verify the counterexample (3.3) over \(k[\epsilon]/(\epsilon^2)\).

Solution. All cohomology spaces are finite-dimensional and occur in finitely many degrees. The long exact cohomology sequence is finite and carries commuting endomorphisms. For any invariant short exact sequence of vector spaces, a splitting makes the endomorphism block triangular and its trace is the sum of its two diagonal traces. Splitting a finite exact sequence into its incoming images and outgoing quotients then makes its alternating trace sum zero. Apply this to the long exact sequence, grouping the \(A,B,C\) terms. By (2.3) the result is (3.5), proving impossibility over a field.

For (3.3), the proposed map on the cone satisfies the chain condition \(0\cdot\epsilon=\epsilon\cdot\epsilon=0\). Its composition with \(i\) is zero. Its composition with \(p\) has component \(\epsilon:C^{-1}\to R[1]^{-1}\), and the identity homotopy from \(C^0\) gives exactly this component. Hence the diagram commutes in the derived category. The cone trace is \((-1)^{-1}\epsilon+0=-\epsilon\), while the first two traces vanish. In the commutative ring \(R\), the additive trace quotient is \(R\), where \(-\epsilon\ne0\). Thus it is a counterexample there.

Exercise 7.3 (medium: a finite coefficient ring). Prove that a bounded complex with finite cohomology modules and finite Tor amplitude over a finite ring is perfect.

Solution. A finite ring is left Noetherian. Construct a bounded-above finite free model from the finitely generated cohomology modules as in Lemma 4.2. At a lower Tor bound \(a\), its cokernel \(C=P^a/\operatorname{im}(d^{a-1})\) has \(\operatorname{Tor}_1(N,C)=0\) for every right module \(N\), so it is flat. It is finitely presented, hence finite projective by the splitting argument in that lemma. Truncating to \(C\to P^{a+1}\to\cdots\) gives the required bounded projective model. No commutativity or finite global dimension of the ring is required.

Exercise 7.4 (medium: the multiplicative group). Compute both Lefschetz numbers of the constant sheaf \(\Lambda=\mathbf Z/\ell^s\) on \(\mathbf G_{m,\mathbf F_q}\).

Solution. Use the compactification \(\mathbf G_m\subset\mathbf P^1\) with complement \(D=\{0,\infty\}\). The localization sequence and the projective-line cohomology used in §5 give

and \(H_c^2(\mathbf G_m,\Lambda)=\Lambda(-1)\), with no other cohomology. The diagonal map is injective, so \(H_c^0=0\), \(H_c^1=\Lambda\). Frobenius is the identity on both rational boundary stalks, hence on this quotient, and acts by \(q\) on \(H_c^2\). The cohomology modules are free, so their alternating trace agrees with the perfect-complex trace: the same cycles-and-boundaries cancellation, or successive splitting of a projective model with projective cohomology, proves this. Thus the global number is \(-1+q\).

The \(q-1\) rational stalks are all trivial rank-one modules, so the local number is also \(q-1\). Both are reduced in \(\Lambda\). This calculation shows why the compact-support degree-one term cannot be omitted.

Exercise 7.5 (hard: a cyclic group without division). Prove (6.11) directly when \(G=\langle s\rangle\) has prime order \(v\), allowing \(v\) to be noninvertible in the coefficient rings.

Solution. Write \(\alpha(s)=s^a\), with \(a\in(\mathbf Z/v)^\times\). Reduce \(P\) to a finite free \(A[G]\)-module by the zero-extension splitting from Theorem 6.2. Write its \(F\)-matrix as in (6.7), with coefficients \(a_{ij,t}\in A\). The coinvariants of its diagonal tensor product have one copy of \(M\) for each \(e_i\). The relation

shows that the induced matrix entry is \(\sum_t a_{ij,t}\,(t^{-1}F)|_M\). Its trace is therefore

If \(a=1\), all \(gF\) are separate \(G\)-conjugacy classes and \(Z_{gF}=G\). Their \(G\)-traces on \(P\) are \(\sum_i a_{ii,g^{-1}}\), by identity-coefficient extraction. Substituting \(t=g^{-1}\) in their sum gives (7.1).

If \(a\ne1\), every \(gF\) lies in one conjugacy class, since conjugation by powers of \(s\) changes its label by all powers \(s^{(a-1)j}\). The centralizer in \(G\) is trivial. In the basis \(s^je_i\), the diagonal terms of \(F\) select \(t=s^{(1-a)j}\), which runs through \(G\) exactly once. Thus \(\operatorname{Tr}_A^{Z_F}(F;P)=\sum_{i,t}a_{ii,t}\). All the operators \(t^{-1}F\) on \(M\) are conjugate, so their traces equal \(\operatorname{Tr}_\Lambda(F;M)\). Equation (7.1) is exactly the one summand in (6.11). Both cases prove the identity. No averaging or cancellation by \(v\) occurred.

Exercise 7.6 (medium: retain the lift). Show by a one-dimensional example that the last factor in (6.11) must be the trace of \(\gamma\), including its \(F\)-action.

Solution. Take \(G=1\), \(A=\Lambda=k\), \(P=k\) with \(F\) acting as \(1\), and \(M=k\) with \(F\) acting as a scalar \(c\ne1\). The coinvariant tensor product is \(k\) with \(F=c\), so its trace is \(c\). The only element above \(1\) is \(F\); the correct right side is \(1\cdot c\). If one replaced \(F\) by its group label \(g=1\) on \(M\), the right side would become \(1\), which is wrong. This verifies the lift convention even before any nontrivial group is introduced.

What this lesson does not prove

• The localization and triangulated-category foundations come from the derived-category course. The bounded projective Hom argument is proved in Theorem 2.2; the injective-embedding and bounded-below resolution providers are specified exactly in §3.
• The finite-coefficient compact-support finiteness input in Theorem 4.3 is the general finiteness statement in Cohomology with compact support, §12. That provider explicitly retains Tag 0GLH as a foundation input. Its Theorem 12.1 proves the bound \(H_c^i(X,B)=0\) for \(i>2\dim X\) for arbitrary torsion sheaves, using its written proper dimension theorem. Theorem 4.4 proves the extension of finite generation to the larger left Noetherian torsion rings used here.
• Compactification, extension by zero, localization, and étale continuity of torsion cohomology under filtered colimits. Their roles and the boundedness required for the projection formula are specified in §4.
• The scheme stratification and constructible-topology compactness inputs of Lemma 4.2b have exact written proofs in Constructible sheaves and extension by zero, Lemmas 4.1 and 5.2. Its coefficient-module and lifting arguments are supplied here for left Noetherian rings.
• Projective-line cohomology, already used in Lesson 1. Its Frobenius convention is proved in the preceding lesson.
• The equality of the two Lefschetz numbers for general \(X_0\) and \(K_0\). This is the subject of the following trace-formula lessons.

References

Stacks, The Trace Formula, sections on traces, derived and filtered derived categories, perfect complexes, Lefschetz numbers and group actions, supplies the baseline. The main result locators are trace well-definedness Tag 03TI, filtered additivity Tag 03TK, the algebraic perfectness criterion Tag 03TO, the projective-curve compact-support argument Tag 03TV, the two Lefschetz numbers Tags 03TX and 03TY, and the group trace identities Tags 03U5, 03U7, 03U9, 03UA, 03UB, 03UC and 03UD. Étale-coefficient definitions and flat models are in Tags 03TQ and 03TT.

These passages appear in AI Integrated Stacks Project, English, The Trace Formula and Étale Cohomology, the AI-integrated edition preserving the upstream locators.

Deligne, Cohomologie étale (SGA \(4\frac12\)), [Rapport], §§4.2–4.9, printed pp. 90–96, develops noncommutative traces, filtered additivity, the finite-Tor criterion and the direct-image perfectness argument. The bounded acyclic-resolution proof of coefficient change in Theorem 4.3 follows the mechanism of §4.9, while §6 supplies a direct integral proof of the coinvariant identity. The historical Lefschetz formalism and its local terms are in Grothendieck–Illusie, SGA 5, Exposé III, “Lefschetz formula,” and Illusie, Exposé III B, “Calculations of local terms.”

The AI Integrated Stacks Project source edition, sections on traces, filtered categories and functors, perfectness, filtrations and Lefschetz numbers, contains the corresponding results. Its omitted filtered and noncommutative arguments are proved above. The finite-degree statement in Lemma 2.3 also appears in the r61 editorial proof supplement, §5. The pseudocoherent criterion is the arbitrary-ring form of the finite-projective truncation argument. Proposition 3.2 specifies the reduced-ring boundary of triangle additivity.

James S. Milne, Lectures on Étale Cohomology, version 2.21, §29, especially pp. 169–175 and Proposition 29.17, discusses perfect complexes and noncommutative traces. The full group-action calculation in §6 works without division by the group order and retains the same lift on both tensor factors. Deligne's Cohomologie étale (SGA \(4\frac12\)), Rapport, §§4.2–4.10, printed pp. 90–96, gives the left Noetherian torsion scope. This lesson's arguments and prose are CC0. Human sources retain their own terms; human Stacks source and modified source supplements retain their GFDL terms.