Lemma 75.12.2 (09TJ)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 75: Derived Categories of Spaces
Section 75.12: The coherator for Noetherian spaces
Lemma 75.12.2 (cite)

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Lemma 75.12.2. Let $S$ be an affine scheme. Let $X$ be a Noetherian algebraic space over $S$ . Every injective object of $\mathit{QCoh}(\mathcal{O}_ X)$ is a direct summand of a filtered colimit $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ of quasi-coherent sheaves of the form

$\mathcal{F}_ i = (Z_ i \to X)_*\mathcal{G}_ i$

where $Z_ i$ is the spectrum of an Artinian ring and $\mathcal{G}_ i$ is a coherent module on $Z_ i$ .

Proof. Choose an affine scheme $U$ and a surjective étale morphism $j : U \to X$ (Properties of Spaces, Lemma 66.6.3). Then $U$ is a Noetherian affine scheme. Choose an injective object $\mathcal{J}'$ of $\mathit{QCoh}(\mathcal{O}_ U)$ such that there exists an injection $\mathcal{J}|_ U \to \mathcal{J}'$ . Then

$\mathcal{J} \to j_*\mathcal{J}'$

is an injective morphism in $\mathit{QCoh}(\mathcal{O}_ X)$ , hence identifies $\mathcal{J}$ as a direct summand of $j_*\mathcal{J}'$ . Thus the result follows from the corresponding result for $\mathcal{J}'$ proved in Lemma 75.12.1. $\square$

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