Lemma 62.15.1. Let $k$ be an algebraically closed field. Let $X$ be a finite type separated scheme over $k$. Let $\Lambda $ be a Noetherian ring. Let $K$ be an object of $D^+_{tors, c}(X_{\acute{e}tale}, \Lambda )$ or of $D_ c(X_{\acute{e}tale}, \Lambda )$ in case $\Lambda $ is torsion. Then $H^ i_ c(X, K)$ is a finite $\Lambda $-module for all $i \in \mathbf{Z}$.

## 62.15 Applications

In this section we give some applications of Theorem 62.14.5.

**Proof.**
Immediate consequence of Theorem 62.14.5 and the definition of compactly supported cohomology in Section 62.12.
$\square$

Proposition 62.15.2. Let $f : X \to S$ be a smooth proper morphism of schemes. Let $\Lambda $ be a Noetherian ring. Let $\mathcal{F}$ be a finite type, locally constant sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$ such that for every geometric point $\overline{x}$ of $X$ the stalk $\mathcal{F}_{\overline{x}}$ is annihilated by an integer $n > 0$ prime to the residue characteristic of $\overline{x}$. Then $R^ if_*\mathcal{F}$ is a finite type, locally constant sheaf of $\Lambda $-modules on $S_{\acute{e}tale}$ for all $i \in \mathbf{Z}$.

**Proof.**
The question is local on $S$ and hence we may assume $S$ is affine. For a point $x$ of $X$ denote $n_ x \geq 1$ the smallest integer annihilating $\mathcal{F}_{\overline{x}}$ for some (equivalently any) geometric point $\overline{x}$ of $X$ lying over $x$. Since $X$ is quasi-compact (being proper over affine) there exists a finite étale covering $\{ U_ j \to X\} _{j = 1, \ldots , m}$ such that $\mathcal{F}|_{U_ j}$ is constant. Since $U_ j \to X$ is open, we conclude that the function $x \mapsto n_ x$ is locally constant and takes finitely many values. Accordingly we obtain a finite decomposition $X = X_1 \amalg \ldots \amalg X_ N$ into open and closed subschemes such that $n_ x = n$ if and only if $x \in X_ n$. Then it suffices to prove the lemma for the induced morphisms $X_ n \to S$ and the restriction of $\mathcal{F}$ to $X_ n$. Thus we may and do assume there exists an integer $n > 0$ such that $\mathcal{F}$ is annihilated by $n$ and such that $n$ is prime to the residue characteristics of all residue fields of $X$.

Since $f$ is smooth and proper the image $f(X) \subset S$ is open and closed. Hence we may replace $S$ by $f(X)$ and assume $f(X) = S$. In particular, we see that we may assume $n$ is invertible in the ring defining the affine scheme $S$.

In this paragraph we reduce to the case where $S$ is Noetherian. Write $S = \mathop{\mathrm{Spec}}(A)$ for some $\mathbf{Z}[1/n]$-algebra $A$. Write $A = \bigcup A_ i$ as the union of its finite type $\mathbf{Z}[1/n]$-subalgebras. We can find an $i$ and a morphism $f_ i : X_ i \to S_ i = \mathop{\mathrm{Spec}}(A_ i)$ of finite type whose base change to $S$ is $f : X \to S$, see Limits, Lemma 32.10.1. After increasing $i$ we may assume $f_ i : X_ i \to S_ i$ is smooth and proper, see Limits, Lemmas 32.13.1 32.8.9, and 32.17.3. By Étale Cohomology, Lemma 59.73.11 we see that there exists an $i$ and a finite type, locally constant sheaf of $\Lambda $-modules $\mathcal{F}_ i$ whose pullback to $X$ is isomorphic to $\mathcal{F}$. As $\mathcal{F}$ is annihilated by $n$, we may replace $\mathcal{F}_ i$ by $\mathop{\mathrm{Ker}}(n : \mathcal{F}_ i \to \mathcal{F}_ i)$ and assume the same thing is true for $\mathcal{F}_ i$. This reduces us to the case discussed in the next paragraph.

Assume we have an integer $n \geq 1$, the base scheme $S$ is Noetherian and lives over $\mathbf{Z}[1/n]$, and $\mathcal{F}$ is $n$-torsion. By Theorem 62.14.5 the sheaves $R^ if_*\mathcal{F}$ are constructible sheaves of $\Lambda $-modules. By Étale Cohomology, Lemma 59.94.3 the specialization maps of $R^ if_*\mathcal{F}$ are always isomorphisms. We conclude by Étale Cohomology, Lemma 59.75.6. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)