The Stacks project

Remark 79.12.6. Let $f : X \to Y$ be a separated, locally quasi-finite morphism of schemes. In this case the sheaf $(X/Y)_{fin}$ is closely related to the sheaf $f_!\mathbf{F}_2$ (insert future reference here) on $Y_{\acute{e}tale}$. Namely, if $V \to Y$ is étale, and $s \in \Gamma (V, f_!\mathbf{F}_2)$, then $s \in \Gamma (V \times _ Y X, \mathbf{F}_2)$ is a section with proper support $Z = \text{Supp}(s)$ over $V$. Since $f$ is also locally quasi-finite we see that the projection $Z \to V$ is actually finite. Since the support of a section of a constant abelian sheaf is open we see that the pair $(V \to Y, \text{Supp}(s))$ satisfies 79.12.0.1. In fact, $f_!\mathbf{F}_2 \cong (X/Y)_{fin}|_{Y_{\acute{e}tale}}$ in this case which also explains the $\mathbf{F}_2$-algebra structure introduced in Remark 79.12.5.