Lemma 74.8.3 (0AHD)—The Stacks project

Loading web-font TeX/Math/Italic

Table of contents
Part 4: Algebraic Spaces
Chapter 74: Descent and Algebraic Spaces
Section 74.8: Descent of finiteness and smoothness properties of morphisms
Lemma 74.8.3 (cite)

numberstags

Lemma 74.8.3. Let $S$ be a scheme. Let

$\xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & B }$

be a commutative diagram of morphisms of algebraic spaces over $S$ . Assume that

$f$ is surjective, and syntomic (resp. smooth, resp. étale),
$p$ is syntomic (resp. smooth, resp. étale).

Then $q$ is syntomic (resp. smooth, resp. étale).

Proof. We deduce this from the analogue for schemes. Namely, the problem is étale local on $B$ and $Y$ (Morphisms of Spaces, Lemmas 67.36.4, 67.37.4, and 67.39.2). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma 67.30.6), we can choose an affine scheme $U$ and an étale morphism $U \to X$ such that the composition $U \to Y$ is surjective. In this case the result follows from Descent, Lemma 35.14.4. $\square$

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