Lemma 96.9.4 (06W9)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 96: Sheaves on Algebraic Stacks
Section 96.9: Restriction
Lemma 96.9.4 (cite)

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Lemma 96.9.4. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\}$ . Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lying over $U = p(x)$ . The equivalence of Lemma 96.9.1 extends to an equivalence of ringed sites $(\mathcal{X}_\tau /x, \mathcal{O}_\mathcal {X}|_ x) \to ((\mathit{Sch}/U)_\tau , \mathcal{O})$ .

Proof. This is immediate from the construction of the structure sheaves. $\square$

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