The Stacks project

Theorem 59.84.7. Let $\Lambda $ be a Noetherian ring, let $k$ be an algebraically closed field, let $X$ be a separated, finite type scheme of dimension $\leq 1$ over $k$, and let $\mathcal{F}$ be a constructible sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$ which is torsion. Then

  1. $H^ q_{\acute{e}tale}(X, \mathcal{F})$ is a finite $\Lambda $-module if $\mathcal{F}$ is torsion prime to $\text{char}(k)$,

  2. $H^ q_{\acute{e}tale}(X, \mathcal{F})$ is a finite $\Lambda $-module if $X$ is proper.

Proof. without further mention. Write $\mathcal{F} = \mathcal{F}_1 \oplus \ldots \oplus \mathcal{F}_ r$ where $\mathcal{F}_ i$ is annihilated by $\ell _ i^{n_ i}$ for some prime $\ell _ i$ and integer $n_ i > 0$. By Lemma 59.84.5 it suffices to prove the theorem for $\mathcal{F}_ i$. Thus we may and do assume that $\ell ^ n$ kills $\mathcal{F}$ for some prime $\ell $ and integer $n > 0$.

Since $\mathcal{F}$ is constructible as a sheaf of $\Lambda $-modules, there is a dense open $U \subset X$ such that $\mathcal{F}|_ U$ is a finite type, locally constant sheaf of $\Lambda $-modules. Since $\dim (X) \leq 1$ we may assume, after shrinking $U$, that $U = U_1 \amalg \ldots \amalg U_ n$ is a disjoint union of irreducible schemes (just remove the closed points which lie in the intersections of $\geq 2$ components of $U$). By Lemma 59.84.3 we reduce to the case $\mathcal{F} = j_!\mathcal{G}$ where $\mathcal{G}$ is a finite type, locally constant sheaf of $\Lambda $-modules on $U$ (and annihilated by $\ell ^ n$).

Since we chose $U = U_1 \amalg \ldots \amalg U_ n$ with $U_ i$ irreducible we have

\[ j_!\mathcal{G} = j_{1!}(\mathcal{G}|_{U_1}) \oplus \ldots \oplus j_{n!}(\mathcal{G}|_{U_ n}) \]

where $j_ i : U_ i \to X$ is the inclusion morphism. The case of $j_{i!}(\mathcal{G}|_{U_ i})$ is handled in Lemma 59.84.6. $\square$

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