Lemma 56.7.2 (0GPK)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 56: Functors and Morphisms
Section 56.7: Functors between categories of coherent modules
Lemma 56.7.2 (cite)

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Lemma 56.7.2. Let $X$ and $Y$ be Noetherian schemes. Let $F : \textit{Coh}(\mathcal{O}_ X) \to \textit{Coh}(\mathcal{O}_ Y)$ be an equivalence of categories. Then there is an isomorphism $f : Y \to X$ and an invertible $\mathcal{O}_ Y$ -module $\mathcal{L}$ such that $F(\mathcal{F}) = f^*\mathcal{F} \otimes \mathcal{L}$ .

Proof. By Lemma 56.7.1 we obtain a unique functor $F' : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Y)$ extending $F$ . The same is true for the quasi-inverse of $F$ and by the uniqueness we conclude that $F'$ is an equivalence. By Proposition 56.6.6 we find an isomorphism $f : Y \to X$ and an invertible $\mathcal{O}_ Y$ -module $\mathcal{L}$ such that $F'(\mathcal{F}) = f^*\mathcal{F} \otimes \mathcal{L}$ . Then $f$ and $\mathcal{L}$ work for $F$ as well. $\square$

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