Lemma 61.3.8 (096L)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.3: Local isomorphisms
Lemma 61.3.8 (cite)

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Lemma 61.3.8. Let $A$ be a ring. Set $X = \mathop{\mathrm{Spec}}(A)$ . The functor

$B \longmapsto \mathop{\mathrm{Spec}}(B)$

from the category of $A$ -algebras $B$ such that $A \to B$ identifies local rings to the category of topological spaces over $X$ is fully faithful.

Proof. This follows from Lemma 61.3.7 and the fact that if $A \to B$ identifies local rings, then the pullback of the structure sheaf of $\mathop{\mathrm{Spec}}(A)$ via $p : \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is equal to the structure sheaf of $\mathop{\mathrm{Spec}}(B)$ . $\square$

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