Lemma 98.25.2 (0GE3)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 98: Artin's Axioms
Section 98.25: Limit preserving functors on Noetherian schemes
Lemma 98.25.2 (cite)

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Lemma 98.25.2. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\}$ . Restricting along the inclusion functor $(\textit{Noetherian}/S)_\tau \to (\mathit{Sch}/S)_\tau$ defines an equivalence of categories between

the category of limit preserving sheaves on $(\mathit{Sch}/S)_\tau$ and
the category of limit preserving sheaves on $(\textit{Noetherian}/S)_\tau$

Proof. Let $F : (\textit{Noetherian}/S)_\tau ^{opp} \to \textit{Sets}$ be a functor which is both limit preserving and a sheaf. By Topologies, Lemmas 34.13.1 and 34.13.3 there exists a unique functor $F' : (\mathit{Sch}/S)_\tau ^{opp} \to \textit{Sets}$ which is limit preserving, a sheaf, and restricts to $F$ . In fact, the construction of $F'$ in Topologies, Lemma 34.13.1 is functorial in $F$ and this construction is a quasi-inverse to restriction. Some details omitted. $\square$

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