Lemma 96.6.2 (06TW)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 96: Sheaves on Algebraic Stacks
Section 96.6: The structure sheaf
Lemma 96.6.2 (cite)

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Lemma 96.6.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$ -morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ . Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\}$ . There is a canonical identification $f^{-1}\mathcal{O}_\mathcal {Y} = \mathcal{O}_\mathcal {X}$ which turns $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau ) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_\tau )$ into a morphism of ringed topoi.

Proof. Denote $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ and $q : \mathcal{Y} \to (\mathit{Sch}/S)_{fppf}$ the structural functors. Then $p = q \circ f$ , hence $p^{-1} = f^{-1} \circ q^{-1}$ by Lemma 96.3.2. Since $\mathcal{O}_\mathcal {X} = p^{-1}\mathcal{O}$ and $\mathcal{O}_\mathcal {Y} = q^{-1}\mathcal{O}$ the result follows. $\square$

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