The Stacks project

Remark 52.18.3. Let $Y$ be a Noetherian scheme and let $Z \subset Y$ be a closed subset. By Lemma 52.18.1 we have

\[ \delta _ Z(y) \leq \min \left\{ k \middle | \begin{matrix} \text{ there exist specializations in }Y \\ y_0 \leftarrow y'_0 \rightarrow y_1 \leftarrow y'_1 \rightarrow \ldots \leftarrow y'_{k - 1} \rightarrow y_ k = y \\ \text{ with }y_0 \in Z\text{ and }y_ i' \leadsto y_ i \text{ immediate} \end{matrix} \right\} \]

We claim that if $Y$ is of finite type over a field, then equality holds. If we ever need this result we will formulate a precise result and prove it here. However, in general if we define $\delta _ Z$ by the right hand side of this inequality, then we don't know if Lemma 52.18.2 remains true.

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