Lemma 99.5.8 (08W7)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 99: Quot and Hilbert Spaces
Section 99.5: The stack of coherent sheaves
Lemma 99.5.8 (cite)

numberstags

$\xymatrix{ X \ar[d] \ar[r]_ i & X' \ar[d] \\ T \ar[r] & T' }$

be a cartesian square of algebraic spaces where $T \to T'$ is a first order thickening. Let $\mathcal{F}'$ be an $\mathcal{O}_{X'}$ -module flat over $T'$ . Set $\mathcal{F} = i^*\mathcal{F}'$ . The following are equivalent

$\mathcal{F}'$ is a quasi-coherent $\mathcal{O}_{X'}$ -module of finite presentation,
$\mathcal{F}'$ is an $\mathcal{O}_{X'}$ -module of finite presentation,
$\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$ -module of finite presentation,
$\mathcal{F}$ is an $\mathcal{O}_ X$ -module of finite presentation,

Proof. Recall that a finitely presented module is quasi-coherent hence the equivalence of (1) and (2) and (3) and (4). The equivalence of (2) and (4) is a special case of Deformation Theory, Lemma 91.11.3. $\square$

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