The Stacks project

Lemma 66.45.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

  1. $f$ is representable and integral (resp. finite),

  2. $f$ is integral (resp. finite),

  3. there exists a scheme $V$ and a surjective ├ętale morphism $V \to Y$ such that $V \times _ Y X \to V$ is integral (resp. finite), and

  4. there exists a Zariski covering $Y = \bigcup Y_ i$ such that each of the morphisms $f^{-1}(Y_ i) \to Y_ i$ is integral (resp. finite).

Proof. It is clear that (1) implies (2) and that (2) implies (3) by taking $V$ to be a disjoint union of affines ├ętale over $Y$, see Properties of Spaces, Lemma 65.6.1. Assume $V \to Y$ is as in (3). Then for every affine open $W$ of $V$ we see that $W \times _ Y X$ is an affine open of $V \times _ Y X$. Hence by Properties of Spaces, Lemma 65.13.1 we conclude that $V \times _ Y X$ is a scheme. Moreover the morphism $V \times _ Y X \to V$ is affine. This means we can apply Spaces, Lemma 64.11.5 because the class of integral (resp. finite) morphisms satisfies all the required properties (see Morphisms, Lemmas 29.44.6 and Descent, Lemmas 35.23.22, 35.23.23, and 35.37.1). The conclusion of applying this lemma is that $f$ is representable and integral (resp. finite), i.e., (1) holds.

The equivalence of (1) and (4) follows from the fact that being integral (resp. finite) is Zariski local on the target (the reference above shows that being integral or finite is in fact fpqc local on the target). $\square$


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