Lemma 66.27.2 (04M5)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.27: Étale localization
Lemma 66.27.2 (cite)

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Lemma 66.27.2. Same notation and assumptions as in Lemma 66.27.1 except that we also assume $U$ and $V$ are schemes. Via the identifications (66.27.0.4) for $U \to X$ and $V \to Y$ the morphism of ringed topoi

$(g_{small}, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}), \mathcal{O}_ U) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (V_{\acute{e}tale}), \mathcal{O}_ V)$

is $2$ -isomorphic to the morphism $(f_{small, s}, f_ s^\sharp )$ constructed in Modules on Sites, Lemma 18.22.3 starting with $(f_{small}, f^\sharp )$ and the map $s : h_ U \to f_{small}^{-1}h_ V$ corresponding to $g$ .

Proof. Note that $(g_{small}, g^\sharp )$ is $2$ -isomorphic as a morphism of ringed topoi to the morphism of ringed topoi associated to the morphism of ringed sites $(g_{spaces, {\acute{e}tale}}, g^\sharp )$ . Hence we conclude by Lemma 66.27.1 and Modules on Sites, Lemma 18.22.4. $\square$

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