Lemma 70.5.14 (0A0R)—The Stacks project

Processing math: 100%

Table of contents
Part 4: Algebraic Spaces
Chapter 70: Limits of Algebraic Spaces
Section 70.5: Descending properties
Lemma 70.5.14 (cite)

numberstags

Lemma 70.5.14. Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$ . Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume

$Y$ quasi-compact and quasi-separated,
$X_ i$ quasi-compact and quasi-separated,
$X \to Y$ affine.

Then $X_ i \to Y$ is affine for $i$ large enough.

Proof. Choose an affine scheme $W$ and a surjective étale morphism $W \to Y$ . Then $X \times _ Y W$ is affine and it suffices to check that $X_ i \times _ Y W$ is affine for some $i$ (Morphisms of Spaces, Lemma 67.20.3). This follows from Lemma 70.5.10. $\square$

Comments (0)

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).

Comments (0)

Post a comment