The Stacks project

Lemma 29.5.3. Let $\mathcal{F}$ be a finite type quasi-coherent module on a scheme $X$. Then

  1. The support of $\mathcal{F}$ is closed.

  2. For $x \in X$ we have

    \[ x \in \text{Supp}(\mathcal{F}) \Leftrightarrow \mathcal{F}_ x \not= 0 \Leftrightarrow \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \kappa (x) \not= 0. \]
  3. For any morphism of schemes $f : Y \to X$ the pullback $f^*\mathcal{F}$ is of finite type as well and we have $\text{Supp}(f^*\mathcal{F}) = f^{-1}(\text{Supp}(\mathcal{F}))$.

Proof. Part (1) is a reformulation of Modules, Lemma 17.9.6. You can also combine Lemma 29.5.1, Properties, Lemma 28.16.1, and Algebra, Lemma 10.40.5 to see this. The first equivalence in (2) is the definition of support, and the second equivalence follows from Nakayama's lemma, see Algebra, Lemma 10.20.1. Let $f : Y \to X$ be a morphism of schemes. Note that $f^*\mathcal{F}$ is of finite type by Modules, Lemma 17.9.2. For the final assertion, let $y \in Y$ with image $x \in X$. Recall that

\[ (f^*\mathcal{F})_ y = \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \mathcal{O}_{Y, y}, \]

see Sheaves, Lemma 6.26.4. Hence $(f^*\mathcal{F})_ y \otimes \kappa (y)$ is nonzero if and only if $\mathcal{F}_ x \otimes \kappa (x)$ is nonzero. By (2) this implies $x \in \text{Supp}(\mathcal{F})$ if and only if $y \in \text{Supp}(f^*\mathcal{F})$, which is the content of assertion (3). $\square$


Comments (0)

There are also:

  • 3 comment(s) on Section 29.5: Supports of modules

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 056J. Beware of the difference between the letter 'O' and the digit '0'.