Definition 29.5.5 (05JV)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.5: Supports of modules
Definition 29.5.5 (cite)

previous lemma

numberstags

Definition 29.5.5. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module of finite type. The scheme theoretic support of $\mathcal{F}$ is the closed subscheme $Z \subset X$ constructed in Lemma 29.5.4.

Comments (2)

Comment #7212 by DatPham on April 28, 2022 at 06:50

I am wondering if there is any reason why the support defined in this way deserves the name "scheme theoretic support", and not the one defined using the Fitting ideal sheaf of $\mathcal{F}$ ? I think the latter bahaves nicer in famillies as its formation commutes with arbitrary base change, which is not the case for the former.

Comment #7327 by Johan on May 04, 2022 at 17:06

This is the definition used most often in the literature.

There are also:

3 comment(s) on Section 29.5: Supports 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 (2)

Post a comment