The Stacks project

Lemma 42.10.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module.

  1. The collection of irreducible components of the support of $\mathcal{F}$ is locally finite.

  2. Let $Z' \subset \text{Supp}(\mathcal{F})$ be an irreducible component and let $\xi \in Z'$ be its generic point. Then

    \[ \text{length}_{\mathcal{O}_{X, \xi }} \mathcal{F}_\xi < \infty \]
  3. If $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k$ and $\xi \in \text{Supp}(\mathcal{F})$ with $\delta (\xi ) = k$, then $\xi $ is a generic point of an irreducible component of $\text{Supp}(\mathcal{F})$.

Proof. By Cohomology of Schemes, Lemma 30.9.7 the support $Z$ of $\mathcal{F}$ is a closed subset of $X$. We may think of $Z$ as a reduced closed subscheme of $X$ (Schemes, Lemma 26.12.4). Hence (1) follows from Divisors, Lemma 31.26.1 applied to $Z$ and (3) follows from Lemma 42.9.1 applied to $Z$.

Let $\xi \in Z'$ be as in (2). In this case for any specialization $\xi ' \leadsto \xi $ in $X$ we have $\mathcal{F}_{\xi '} = 0$. Recall that the non-maximal primes of $\mathcal{O}_{X, \xi }$ correspond to the points of $X$ specializing to $\xi $ (Schemes, Lemma 26.13.2). Hence $\mathcal{F}_\xi $ is a finite $\mathcal{O}_{X, \xi }$-module whose support is $\{ \mathfrak m_\xi \} $. Hence it has finite length by Algebra, Lemma 10.62.3. $\square$

Comments (2)

Comment #8834 by Rankeya on

In the statement of (3) should be ?

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Your email address will not be published.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 02QW. Beware of the difference between the letter 'O' and the digit '0'.