Lemma 71.10.8. 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 $.

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