Lemma 76.44.6 (09RY)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.44: Regular immersions
Lemma 76.44.6 (cite)

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Lemma 76.44.6. Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$ . Then $i$ is a quasi-regular immersion if and only if the following conditions are satisfied

$i$ is locally of finite presentation,
the conormal sheaf $\mathcal{C}_{Z/X}$ is finite locally free, and
the map (76.6.1.2) is an isomorphism.

Proof. Follows from the case of schemes (Divisors, Lemma 31.21.5) via étale localization (use Definition 76.44.2 and Lemma 76.6.2). $\square$

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