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

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. $\square$

## 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Your email address will not be published.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)