Lemma 18.40.5 (04H7)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.40: Locally ringed topoi
Lemma 18.40.5 (cite)

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Lemma 18.40.5. Being a locally ringed site is an intrinsic property. More precisely,

if $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is a morphism of topoi and $(\mathcal{C}, \mathcal{O})$ is a locally ringed site, then $(\mathcal{C}', f^{-1}\mathcal{O})$ is a locally ringed site, and
if $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is an equivalence of ringed topoi, then $(\mathcal{C}, \mathcal{O})$ is locally ringed if and only if $(\mathcal{C}', \mathcal{O}')$ is locally ringed.

Proof. It is clear that (2) follows from (1). To prove (1) note that as $f^{-1}$ is exact we have $f^{-1}* = *$, $f^{-1}\emptyset ^\# = \emptyset ^\# $, and $f^{-1}$ commutes with products, equalizers and transforms isomorphisms and surjections into isomorphisms and surjections. Thus $f^{-1}$ transforms the isomorphism (18.40.2.1) into its analogue for $f^{-1}\mathcal{O}$ and transforms the surjection of Lemma 18.40.1 part (3) into the corresponding surjection for $f^{-1}\mathcal{O}$. $\square$

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