The Stacks project

Lemma 66.10.8. Let $k$ be a field. Let $X$ be an algebraic space over $k$. The following are equivalent

  1. $X$ is locally quasi-finite over $k$,

  2. $X$ is locally of finite type over $k$ and has dimension $0$,

  3. $X$ is a scheme and is locally quasi-finite over $k$,

  4. $X$ is a scheme and is locally of finite type over $k$ and has dimension $0$, and

  5. $X$ 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 61.34.6 we see that (1) and (2) are equivalent. Hence it follows from Lemma 66.9.1 (and trivial implications) that (1) – (4) are equivalent. Finally, Varieties, Lemma 32.20.2 shows that (1) – (4) are equivalent with (5). $\square$


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