Lemma 72.10.8 (06S0)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 72: Algebraic Spaces over Fields
Section 72.10: Schematic locus and field extension
Lemma 72.10.8 (cite)

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Lemma 72.10.8. Let $k$ be a field. Let $X$ be an algebraic space over $k$ . The following are equivalent

$X$ is locally quasi-finite over $k$ ,
$X$ is locally of finite type over $k$ and has dimension $0$ ,
$X$ is a scheme and is locally quasi-finite over $k$ ,
$X$ is a scheme and is locally of finite type over $k$ and has dimension $0$ , and
$X$ is a disjoint union of spectra of Artinian local $k$ -algebras $A$ over $k$ with $\dim _ k(A) < \infty$ .

Proof. Because we are over a field relative dimension of $X/k$ is the same as the dimension of $X$ . Hence by Morphisms of Spaces, Lemma 67.34.6 we see that (1) and (2) are equivalent. Hence it follows from Lemma 72.9.1 (and trivial implications) that (1) – (4) are equivalent. Finally, Varieties, Lemma 33.20.2 shows that (1) – (4) are equivalent with (5). $\square$

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