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 $\text{Supp}(\mathcal{F})$.

## Comments (2)

Comment #8834 by Rankeya on

Comment #9256 by Stacks project on