The Stacks project

Lemma 74.11.16. The property $\mathcal{P}(f) =$“$f$ is affine” is fpqc local on the base.

Proof. We will use Lemma 74.10.4 to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 67.20.3. Let $Z' \to Z$ be a surjective flat morphism of affine schemes over $S$. Let $f : X \to Z$ be a morphism of algebraic spaces, and assume that the base change $f' : Z' \times _ Z X \to Z'$ is affine. Let $X'$ be a scheme representing $Z' \times _ Z X$. We obtain a canonical isomorphism

\[ \varphi : X' \times _ Z Z' \longrightarrow Z' \times _ Z X' \]

since both schemes represent the algebraic space $Z' \times _ Z Z' \times _ Z X$. This is a descent datum for $X'/Z'/Z$, see Descent, Definition 35.34.1 (verification omitted, compare with Descent, Lemma 35.39.1). Since $X' \to Z'$ is affine this descent datum is effective, see Descent, Lemma 35.37.1. Thus there exists a scheme $Y \to Z$ over $Z$ and an isomorphism $\psi : Z' \times _ Z Y \to X'$ compatible with descent data. Of course $Y \to Z$ is affine (by construction or by Descent, Lemma 35.23.18). Note that $\mathcal{Y} = \{ Z' \times _ Z Y \to Y\} $ is a fpqc covering, and interpreting $\psi $ as an element of $X(Z' \times _ Z Y)$ we see that $\psi \in \check{H}^0(\mathcal{Y}, X)$. By the sheaf condition for $X$ with respect to this covering (see Properties of Spaces, Proposition 66.17.1) we obtain a morphism $Y \to X$. By construction the base change of this to $Z'$ is an isomorphism, hence an isomorphism by Lemma 74.11.15. This proves that $X$ is representable by an affine scheme and we win. $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 041Z. Beware of the difference between the letter 'O' and the digit '0'.