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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