Lemma 59.70.7. Let $j : U \to X$ be finite and étale. Then the map $j_! \to j_*$ of Lemma 59.70.6 is an isomorphism on abelian sheaves and sheaves of $\Lambda $-modules.

**Proof.**
It suffices to check $j_!\mathcal{F} \to j_*\mathcal{F}$ is an isomorphism étale locally on $X$. Thus we may assume $U \to X$ is a finite disjoint union of isomorphisms, see Étale Morphisms, Lemma 41.18.3. We omit the proof in this case.
$\square$

## 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Your email address will not be published.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)