Lemma 35.7.1 (05AZ)—The Stacks project

Loading web-font TeX/Math/Italic

Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.7: Descent of finiteness properties of modules
Lemma 35.7.1 (cite)

next lemma

numberstags

history
statistics
5 tags refer to this tag

Lemma 35.7.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is a finite type $\mathcal{O}_{X_ i}$ -module. Then $\mathcal{F}$ is a finite type $\mathcal{O}_ X$ -module.

Proof. Omitted. For the affine case, see Algebra, Lemma 10.83.2. $\square$

Comments (0)

There are also:

2 comment(s) on Section 35.7: Descent of finiteness properties of modules

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