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.

## 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.

**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:

$f$ is representable and quasi-affine,

$f$ is quasi-affine,

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

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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)