The Stacks project

Lemma 61.16.2. Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a sheaf of sets on $X_{pro\text{-}\acute{e}tale}$. If $W$ is an affine weakly contractible object of $X_{pro\text{-}\acute{e}tale}$, then

\[ f_{small}^{-1}\mathcal{F}(W) = \mathop{\mathrm{colim}}\nolimits _{W \to V} \mathcal{F}(V) \]

where the colimit is over morphisms $W \to V$ over $Y$ with $V \in Y_{pro\text{-}\acute{e}tale}$.

Proof. Recall that $f_{small}^{-1}\mathcal{F}$ is the sheaf associated to the presheaf

\[ u_ p\mathcal{F} : U \mapsto \mathop{\mathrm{colim}}\nolimits _{U \to V} \mathcal{F}(V) \]

on $X_{\acute{e}tale}$, see Sites, Sections 7.14 and 7.13; we've suppressed from the notation that the colimit is over the opposite of the category $\{ U \to V, V \in Y_{pro\text{-}\acute{e}tale}\} $. By Lemma 61.16.1 it suffices to prove that $u_ p\mathcal{F}$ sends finite disjoint unions to products. Suppose that $U = U_1 \amalg U_2$ is a disjoint union of open and closed subschemes. There is a functor

\[ \{ U_1 \to V_1\} \times \{ U_2 \to V_2\} \longrightarrow \{ U \to V\} ,\quad (U_1 \to V_1, U_2 \to V_2) \longmapsto (U \to V_1 \amalg V_2) \]

which is initial (Categories, Definition 4.17.3). Hence the corresponding functor on opposite categories is cofinal and by Categories, Lemma 4.17.2 we see that $u_ p\mathcal{F}$ on $U$ is the colimit of the values $\mathcal{F}(V_1 \amalg V_2)$ over the product category. Since $\mathcal{F}$ is a sheaf it sends disjoint unions to products and we conclude $u_ p\mathcal{F}$ does too. $\square$

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