Lemma 77.6.4 (0CW4)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 77: Flatness on Algebraic Spaces
Section 77.6: A criterion for purity
Lemma 77.6.4 (cite)

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Lemma 77.6.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ . Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Let $y \in |Y|$ . Assume

$f$ is decent and of finite type,
$\mathcal{F}$ is of finite type,
$\mathcal{F}$ is flat over $Y$ at all points lying over $y$ , and
$\mathcal{F}$ is pure above $y$ .

Then $\mathcal{F}$ is universally pure above $y$ .

Proof. Consider the morphism $\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}) \to Y$ . This is a flat morphism from the spectrum of a strictly henselian local ring which maps the closed point to $y$ . By Lemma 77.3.4 we reduce to the case described in the next paragraph.

Assume $Y$ is the spectrum of a strictly henselian local ring $R$ with closed point $y$ . By Lemma 77.4.6 there exists an étale morphism $g : X' \to X$ with $g(|X'|) \supset |X_ y|$ , with $X'$ affine, and with $\Gamma (X', g^*\mathcal{F})$ a free $R$ -module. Then $g^*\mathcal{F}$ is universally pure relative to $Y$ , see More on Flatness, Lemma 38.17.4. Hence it suffices to prove that $g(|X'|)$ contains $\text{Ass}_{X/Y}(\mathcal{F})$ by Lemma 77.6.3 part (1). This in turn follows from Lemma 77.6.3 part (2). $\square$

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