Lemma 62.14.1. Let $f : X \to Y$ be a morphism of schemes which is locally quasi-finite and of finite presentation. The functor $f_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda )$ of Lemma 62.7.1 sends $D_ c(X_{\acute{e}tale}, \Lambda )$ into $D_ c(Y_{\acute{e}tale}, \Lambda )$.

**Proof.**
Since the functor $f_!$ is exact, it suffices to show that $f_!\mathcal{F}$ is constructible for any constructible sheaf $\mathcal{F}$ of $\Lambda $-modules on $X_{\acute{e}tale}$. The question is local on $Y$ and hence we may and do assume $Y$ is affine. Then $X$ is quasi-compact and quasi-separated, see Morphisms, Definition 29.21.1. Say $X = \bigcup _{i = 1, \ldots , n} X_ i$ is a finite affine open covering. By Lemma 62.4.7 we see that it suffices to show that $f_{i, !}\mathcal{F}|_{X_ i}$ and $f_{ii', !}\mathcal{F}|_{X_ i \cap X_{i'}}$ are constructible where $f_ i : X_ i \to Y$ and $f_{ii'} : X_ i \cap X_{i'} \to Y$ are the restrictions of $f$. Since $X_ i$ and $X_ i \cap X_{i'}$ are quasi-compact and separated this means we may assume $f$ is separated. By Zariski's main theorem (in the form of More on Morphisms, Lemma 37.43.4) we can choose a factorization $f = g \circ j$ where $j : X \to X'$ is an open immersion and $g : X' \to Y$ is finite and of finite presentation. Then $f_! = g_! \circ j_!$ by Lemma 62.3.13. By Étale Cohomology, Lemma 59.73.1 we see that $j_!\mathcal{F}$ is constructible on $X'$. The morphism $g$ is finite hence $g_! = g_*$ by Lemma 62.3.4. Thus $f_!\mathcal{F} = g_!j_!\mathcal{F} = g_*j_!\mathcal{F}$ is constructible by Étale Cohomology, Lemma 59.73.9.
$\square$

## 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Your email address will not be published.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)