The Stacks project

Lemma 63.14.3. Let $Y$ be a Noetherian affine scheme of finite dimension. Let $\Lambda $ be a Noetherian ring which is torsion. Let $\mathcal{F}$ be a finite type, locally constant sheaf of $\Lambda $-modules on an open subscheme $U \subset \mathbf{A}^1_ Y$. Then $Rf_!\mathcal{F}$ has constructible cohomology sheaves where $f : U \to Y$ is the structure morphism.

Proof. We may decompose $\Lambda $ as a product $\Lambda = \Lambda _1 \times \ldots \times \Lambda _ r$ where $\Lambda _ i$ is $\ell _ i$-primary for some prime $\ell _ i$. Thus we may assume there exists a prime $\ell $ and an integer $n > 0$ such that $\ell ^ n$ annihilates $\Lambda $ (and hence $\mathcal{F}$).

Since $U$ is Noetherian, we see that $U$ has finitely many connected components. Thus we may assume $U$ is connected. Let $g : U' \to U$ be the finite étale covering constructed in Étale Cohomology, Lemma 59.66.4. The discussion in Étale Cohomology, Section 59.66 gives maps

\[ \mathcal{F} \to g_*g^{-1}\mathcal{F} \to \mathcal{F} \]

whose composition is an isomorphism. Hence it suffices to prove the result for $g_*g^{-1}\mathcal{F}$. On the other hand, we have $Rf_!g_*g^{-1}\mathcal{F} = R(f \circ g)_!g^{-1}\mathcal{F}$ by Lemma 63.9.2. Since $g^{-1}\mathcal{F}$ has a finite filtration by constant sheaves of $\Lambda $-modules of the form $\underline{M}$ for some finite $\Lambda $-module $M$ (by our choice of $g$) this reduces us to the case proved in Lemma 63.14.2. $\square$

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