Lemma 42.10.1 (02QW)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.10: Cycle associated to a coherent sheaf
Lemma 42.10.1 (cite)

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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.

The collection of irreducible components of the support of $\mathcal{F}$ is locally finite.
Let $Z' \subset \text{Supp}(\mathcal{F})$ be an irreducible component and let $\xi \in Z'$ be its generic point. Then

\[ \text{length}_{\mathcal{O}_{X, \xi }} \mathcal{F}_\xi < \infty \]
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 February 28, 2024 at 17:15

In the statement of (3) should $Z$ be $Supp(\mathcal{F})$ ?

Comment #9256 by Stacks project on May 17, 2024 at 14:39

Thanks and fixed here.

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