Lemma 28.10.3 (02IZ)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 28: Properties of Schemes
Section 28.10: Dimension
Lemma 28.10.3 (cite)

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Lemma 28.10.3. Let $X$ be a scheme. Let $Y \subset X$ be an irreducible closed subset. Let $\xi \in Y$ be the generic point. Then

\[ \text{codim}(Y, X) = \dim (\mathcal{O}_{X, \xi }) \]

where the codimension is as defined in Topology, Definition 5.11.1.

Proof. By Topology, Lemma 5.11.2 we may replace $X$ by an affine open neighbourhood of $\xi $. In this case the result follows easily from Algebra, Lemma 10.26.3. $\square$

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