## 66.21 Quasi-affine morphisms

We have already defined in Section 66.3 what it means for a representable morphism of algebraic spaces to be quasi-affine.

Lemma 66.21.1. Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is quasi-affine (in the sense of Section 66.3) if and only if for all affine schemes $Z$ and morphisms $Z \to Y$ the scheme $X \times _ Y Z$ is quasi-affine.

Proof. This follows directly from the definition of a quasi-affine morphism of schemes (Morphisms, Definition 29.13.1). $\square$

This clears the way for the following definition.

Definition 66.21.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is quasi-affine if for every affine scheme $Z$ and morphism $Z \to Y$ the algebraic space $X \times _ Y Z$ is representable by a quasi-affine scheme.

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 étale 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 étale 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). $\square$

Lemma 66.21.4. The composition of quasi-affine morphisms is quasi-affine.

Proof. Omitted. $\square$

Lemma 66.21.5. The base change of a quasi-affine morphism is quasi-affine.

Proof. Omitted. $\square$

Lemma 66.21.6. Let $S$ be a scheme. A quasi-compact and quasi-separated morphism of algebraic spaces $f : Y \to X$ is quasi-affine if and only if the canonical factorization $Y \to \underline{\mathop{\mathrm{Spec}}}_ X(f_*\mathcal{O}_ Y)$ (Remark 66.20.9) is an open immersion.

Proof. Let $U \to X$ be a surjective morphism where $U$ is a scheme. Since we may check whether $f$ is quasi-affine after base change to $U$ (Lemma 66.21.3), since $f_*\mathcal{O}_ Y|_ V$ is equal to $(Y \times _ X U \to U)_*\mathcal{O}_{Y \times _ X U}$ (Properties of Spaces, Lemma 65.26.2), and since formation of relative spectrum commutes with base change (Lemma 66.20.7), we see that the assertion reduces to the case that $X$ is a scheme. If $X$ is a scheme and either $f$ is quasi-affine or $Y \to \underline{\mathop{\mathrm{Spec}}}_ X(f_*\mathcal{O}_ Y)$ is an open immersion, then $Y$ is a scheme as well. Thus we reduce to Morphisms, Lemma 29.13.3. $\square$

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