Lemma 42.9.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme.

Let $Z' \subset Z$ be an irreducible component and let $\xi \in Z'$ be its generic point. Then

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

## Comments (4)

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