Theorem 76.24.1 (0APR)—The Stacks project

Processing math: 100%

Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.24: Flatness over a Noetherian base
Theorem 76.24.1 (cite)

next lemma

numberstags

Theorem 76.24.1. Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be morphisms of algebraic spaces over $S$ . Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Assume

$X$ , $Y$ , $Z$ locally Noetherian, and
$\mathcal{F}$ a coherent $\mathcal{O}_ X$ -module.

Let $x \in |X|$ and let $y \in |Y|$ and $z \in |Z|$ be the images of $x$ . If $\mathcal{F}_{\overline{x}} \not= 0$ , then the following are equivalent:

$\mathcal{F}$ is flat over $Z$ at $x$ and the restriction of $\mathcal{F}$ to its fibre over $z$ is flat at $x$ over the fibre of $Y$ over $z$ , and
$Y$ is flat over $Z$ at $y$ and $\mathcal{F}$ is flat over $Y$ at $x$ .

Proof. Choose a diagram as in Lemma 76.23.1 part (3). It follows from the definitions that this reduces to the corresponding theorem for the morphisms of schemes $U \to V \to W$ , the quasi-coherent sheaf $a^*\mathcal{F}$ , and the point $u \in U$ . Thus the theorem follows from the corresponding result for schemes which is More on Morphisms, Theorem 37.16.1. $\square$

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