The Stacks project

Lemma 62.3.1. Let $f : X \to Y$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. The rule

\[ Y_{\acute{e}tale}\longrightarrow \textit{Ab},\quad V \longmapsto \{ s \in f_*\mathcal{F}(V) = \mathcal{F}(X_ V) \mid \text{Supp}(s) \subset X_ V \text{ is proper over }V\} \]

is an abelian subsheaf of $f_*\mathcal{F}$.

Proof. Recall that the support of a section is closed (Étale Cohomology, Lemma 59.31.4) hence the material in Cohomology of Schemes, Section 30.26 applies. By the lemma above and Cohomology of Schemes, Lemma 30.26.6 we find that our subset of $f_*\mathcal{F}(V)$ is a subgroup. By Cohomology of Schemes, Lemma 30.26.4 we see that our rule defines a sub presheaf. Finally, suppose that we have $s \in f_*\mathcal{F}(V)$ and an étale covering $\{ V_ i \to V\} $ such that $s|_{V_ i}$ has support proper over $V_ i$. Observe that the support of $s|_{V_ i}$ is the inverse image of the support of $s|_ V$ (use the characterization of the support in terms of stalks and Étale Cohomology, Lemma 59.36.2). Whence the support of $s$ is proper over $V$ by Descent, Lemma 35.25.5. This proves that our rule satisfies the sheaf condition. $\square$


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