Theorem 64.20.1 (Finite Coefficients). Let $X$ be a scheme of finite type over a finite field $k$. Let $\Lambda $ be a finite ring of order prime to the characteristic of $k$ and $\mathcal{F}$ a constructible flat $\Lambda $-module on $X_{\acute{e}tale}$. Then

## 64.20 Cohomological interpretation

This is how Grothendieck interpreted the $L$-function.

**Proof.**
Omitted.
$\square$

Thus far, we don't even know whether each cohomology group $H^ i_ c(X_{\bar k}, \mathcal{F})$ is free.

Theorem 64.20.2 (Adic sheaves). Let $X$ be a scheme of finite type over a finite field $k$, and $\mathcal{F}$ a $\mathbf{Q}_\ell $-sheaf on $X$. Then

**Proof.**
This is sketched below.
$\square$

Remark 64.20.3. Since we have only developed some theory of traces and not of determinants, Theorem 64.20.1 is harder to prove than Theorem 64.20.2. We will only prove the latter, for the former see [SGA4.5]. Observe also that there is no version of this theorem more general for $\mathbf{Z}_\ell $ coefficients since there is no $\ell $-torsion.

We reduce the proof of Theorem 64.20.2 to a trace formula. Since $\mathbf{Q}_\ell $ has characteristic 0, it suffices to prove the equality after taking logarithmic derivatives. More precisely, we apply $T\frac{d}{dT} \log $ to both sides. We have on the one hand

where the last equality results from the formula

which holds for any commutative ring $\Lambda $ and any endomorphism $f$ of a finite projective $\Lambda $-module $M$. On the other hand, we have

by the same formula again. Now, comparing powers of $T$ and using the Mobius inversion formula, we see that Theorem 64.20.2 is a consequence of the following equality

Writing $k_ n$ for the degree $n$ extension of $k$, $X_ n = X \times _{\mathop{\mathrm{Spec}}k} \mathop{\mathrm{Spec}}(k_ n)$ and $_ n\mathcal{F} = \mathcal{F}|_{X_ n}$, this boils down to

which is a consequence of Theorem 64.20.5.

Theorem 64.20.4. Let $X/k$ be as above, let $\Lambda $ be a finite ring with $\# \Lambda \in k^*$ and $K\in D_{ctf}(X, \Lambda )$. Then $R\Gamma _ c(X_{\bar k}, K)\in D_{perf}(\Lambda )$ and

**Proof.**
Note that we have already proved this (REFERENCE) when $\dim X \leq 1$. The general case follows easily from that case together with the proper base change theorem.
$\square$

Theorem 64.20.5. Let $X$ be a separated scheme of finite type over a finite field $k$ and $\mathcal{F}$ be a $\mathbf{Q}_\ell $-sheaf on $X$. Then $\dim _{\mathbf{Q}_\ell }H_ c^ i(X_{\bar k}, \mathcal{F})$ is finite for all $i$, and is nonzero for $0\leq i \leq 2 \dim X$ only. Furthermore, we have

**Proof.**
We explain how to deduce this from Theorem 64.20.4. We first use some étale cohomology arguments to reduce the proof to an algebraic statement which we subsequently prove.

Let $\mathcal{F}$ be as in the theorem. We can write $\mathcal{F}$ as $\mathcal{F}'\otimes \mathbf{Q}_\ell $ where $\mathcal{F}' = \left\{ \mathcal{F}'_ n\right\} $ is a $\mathbf{Z}_\ell $-sheaf without torsion, i.e., $\ell : \mathcal{F}'\to \mathcal{F}'$ has trivial kernel in the category of $\mathbf{Z}_\ell $-sheaves. Then each $\mathcal{F}_ n'$ is a flat constructible $\mathbf{Z}/\ell ^ n\mathbf{Z}$-module on $X_{\acute{e}tale}$, so $\mathcal{F}'_ n \in D_{ctf}(X, \mathbf{Z}/\ell ^ n\mathbf{Z})$ and $\mathcal{F}_{n+1}' \otimes ^{\mathbf{L}}_{\mathbf{Z}/\ell ^{n+1}\mathbf{Z}} \mathbf{Z}/\ell ^ n\mathbf{Z} = \mathcal{F}_ n'$. Note that the last equality holds also for standard (non-derived) tensor product, since $\mathcal{F}'_ n$ is flat (it is the same equality). Therefore,

the complex $K_ n = R\Gamma _ c\left(X_{\bar k}, \mathcal{F}_ n'\right)$ is perfect, and it is endowed with an endomorphism $\pi _ n : K_ n\to K_ n$ in $D(\mathbf{Z}/\ell ^ n\mathbf{Z})$,

there are identifications

\[ K_{n+1} \otimes ^{\mathbf{L}}_{\mathbf{Z}/\ell ^{n+1}\mathbf{Z}} \mathbf{Z}/\ell ^ n\mathbf{Z} = K_ n \]in $D_{perf}(\mathbf{Z}/\ell ^ n\mathbf{Z})$, compatible with the endomorphisms $\pi _{n+1}$ and $\pi _ n$ (see [Rapport 4.12, SGA4.5]),

the equality $\text{Tr}\left(\pi _ X^* |_{K_ n}\right) = \sum _{x\in X(k)} \text{Tr}\left(\pi _ x |_{(\mathcal{F}'_ n)_{\bar x}}\right)$ holds, and

for each $x\in X(k)$, the elements $\text{Tr}(\pi _ x |_{\mathcal{F}'_{n, \bar x}}) \in \mathbf{Z}/\ell ^ n\mathbf{Z}$ form an element of $\mathbf{Z}_\ell $ which is equal to $\text{Tr}(\pi _ x |_{\mathcal{F}_{\bar x}}) \in \mathbf{Q}_\ell $.

It thus suffices to prove the following algebra lemma. $\square$

Lemma 64.20.6. Suppose we have $K_ n\in D_{perf}(\mathbf{Z}/\ell ^ n\mathbf{Z})$, $\pi _ n : K_ n\to K_ n$ and isomorphisms $\varphi _ n : K_{n+1} \otimes ^\mathbf {L}_{\mathbf{Z}/\ell ^{n+1}\mathbf{Z}} \mathbf{Z}/\ell ^ n\mathbf{Z} \to K_ n$ compatible with $\pi _{n+1}$ and $\pi _ n$. Then

the elements $t_ n = \text{Tr}(\pi _ n |_{K_ n})\in \mathbf{Z}/\ell ^ n\mathbf{Z}$ form an element $t_\infty = \{ t_ n\} $ of $\mathbf{Z}_\ell $,

the $\mathbf{Z}_\ell $-module $H_\infty ^ i = \mathop{\mathrm{lim}}\nolimits _ n H^ i(k_ n)$ is finite and is nonzero for finitely many $i$ only, and

the operators $H^ i(\pi _ n): H^ i(K_ n)\to H^ i(K_ n)$ are compatible and define $\pi _\infty ^ i : H_\infty ^ i\to H_\infty ^ i$ satisfying

\[ \sum (-1)^ i \text{Tr}( \pi _\infty ^ i |_{H_\infty ^ i \otimes _{\mathbf{Z}_\ell }\mathbf{Q}_\ell }) = t_\infty . \]

**Proof.**
Since $\mathbf{Z}/\ell ^ n\mathbf{Z}$ is a local ring and $K_ n$ is perfect, each $K_ n$ can be represented by a finite complex $K_ n^\bullet $ of finite free $\mathbf{Z}/\ell ^ n \mathbf{Z}$-modules such that the map $K_ n^ p \to K_ n^{p+1}$ has image contained in $\ell K_ n^{p+1}$. It is a fact that such a complex is unique up to isomorphism. Moreover $\pi _ n$ can be represented by a morphism of complexes $\pi _ n^\bullet : K_ n^\bullet \to K_ n^\bullet $ (which is unique up to homotopy). By the same token the isomorphism $\varphi _ n : K_{n+1} \otimes _{\mathbf{Z}/\ell ^{n+1}\mathbf{Z}}^{\mathbf{L}} \mathbf{Z}/\ell ^ n\mathbf{Z}\to K_ n$ is represented by a map of complexes

In fact, $\varphi _ n^\bullet $ is an isomorphism of complexes, thus we see that

there exist $a, b\in \mathbf{Z}$ independent of $n$ such that $K_ n^ i = 0$ for all $i\notin [a, b]$, and

the rank of $K_ n^ i$ is independent of $n$.

Therefore, the module $K_\infty ^ i = \mathop{\mathrm{lim}}\nolimits _ n \{ K_ n^ i, \varphi _ n^ i\} $ is a finite free $\mathbf{Z}_\ell $-module and $K_\infty ^\bullet $ is a finite complex of finite free $\mathbf{Z}_\ell $-modules. By induction on the number of nonzero terms, one can prove that $H^ i\left(K_\infty ^\bullet \right) = \mathop{\mathrm{lim}}\nolimits _ n H^ i\left(K_ n^\bullet \right)$ (this is not true for unbounded complexes). We conclude that $H_\infty ^ i = H^ i\left(K_\infty ^\bullet \right)$ is a finite $\mathbf{Z}_\ell $-module. This proves *ii*. To prove the remainder of the lemma, we need to overcome the possible noncommutativity of the diagrams

However, this diagram does commute in the derived category, hence it commutes up to homotopy. We inductively replace $\pi _ n^\bullet $ for $n\geq 2$ by homotopic maps of complexes making these diagrams commute. Namely, if $h^ i : K_{n+1}^ i \to K_ n^{i-1}$ is a homotopy, i.e.,

then we choose $\tilde h^ i : K_{n+1}^ i\to K_{n+1}^{i-1}$ lifting $h^ i$. This is possible because $K_{n+1}^ i$ free and $K_{n+1}^{i-1}\to K_ n^{i-1}$ is surjective. Then replace $\pi _ n^\bullet $ by $\tilde\pi _ n^\bullet $ defined by

With this choice of $\{ \pi _ n^\bullet \} $, the above diagrams commute, and the maps fit together to define an endomorphism $\pi _\infty ^\bullet = \mathop{\mathrm{lim}}\nolimits _ n\pi _ n^\bullet $ of $K_\infty ^\bullet $. Then part *i* is clear: the elements $t_ n = \sum (-1)^ i \text{Tr}\left(\pi _ n^ i |_{K_ n^ i}\right)$ fit into an element $t_\infty $ of $\mathbf{Z}_\ell $. Moreover

where the last equality follows from the fact that $\mathbf{Q}_\ell $ is a field, so the complex $K_\infty ^\bullet \otimes \mathbf{Q}_\ell $ is quasi-isomorphic to its cohomology $H^ i(K_\infty ^\bullet \otimes \mathbf{Q}_\ell )$. The latter is also equal to $H^ i(K_\infty ^\bullet )\otimes _{\mathbf{Z}}\mathbf{Q}_\ell = H_\infty ^ i \otimes \mathbf{Q}_\ell $, which finishes the proof of the lemma, and also that of Theorem 64.20.5. $\square$

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