The Stacks project

Lemma 66.21.3. 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 quasi-affine,

  2. $f$ is quasi-affine,

  3. there exists a scheme $V$ and a surjective ├ętale morphism $V \to Y$ such that $V \times _ Y X \to V$ is quasi-affine, 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 quasi-affine.

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 a quasi-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 quasi-affine. This means we can apply Spaces, Lemma 64.11.5 because the class of quasi-affine morphisms satisfies all the required properties (see Morphisms, Lemmas 29.13.5 and Descent, Lemmas 35.23.20 and 35.38.1). The conclusion of applying this lemma is that $f$ is representable and quasi-affine, i.e., (1) holds.

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

Comments (1)

Comment #979 by on

Suggested slogan: Morphisms of algebraic spaces that are locally quasi-affine are quasi-affine and representable.

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