The Stacks project

Proposition 61.15.2. 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 32.16.3. By Étale Cohomology, Lemma 58.73.11 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 61.14.5 the sheaves $R^ if_*\mathcal{F}$ are constructible sheaves of $\Lambda $-modules. By Étale Cohomology, Lemma 58.93.3 the specialization maps of $R^ if_*\mathcal{F}$ are always isomorphisms. We conclude by Étale Cohomology, Lemma 58.75.6. $\square$


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