Lemma 29.11.5 (01SA)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.11: Affine morphisms
Lemma 29.11.5 (cite)

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Lemma 29.11.5. Let $S$ be a scheme. There is an anti-equivalence of categories

$\begin{matrix} \text{Schemes affine} \\ \text{over }S \end{matrix} \longleftrightarrow \begin{matrix} \text{quasi-coherent sheaves} \\ \text{of }\mathcal{O}_ S\text{-algebras} \end{matrix}$

which associates to $f : X \to S$ the sheaf $f_*\mathcal{O}_ X$ . Moreover, this equivalence is compatible with arbitrary base change.

Proof. The functor from right to left is given by $\underline{\mathop{\mathrm{Spec}}}_ S$ . The two functors are mutually inverse by Lemma 29.11.3 and Constructions, Lemma 27.4.6 part (3). The final statement is Constructions, Lemma 27.4.6 part (2). $\square$

Comments (2)

Comment #8440 by Elías Guisado on May 29, 2023 at 10:31

Minor style typo: in the statement perhaps one would want to enclose the name of the categories between curly braces to fit the style in 29.11.6.

2nd proof (maybe worth of mention?): The functor $\underline{\text{Spec}}_S$ is fully faithful by remark (ii) of https://stacks.math.columbia.edu/tag/01LQ#comment-8438 , and it is essentially surjective by 29.11.3.

Comment #9063 by Stacks project on May 06, 2024 at 12:53

Going to leave as is for now.

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