The Stacks project

Lemma 33.5.2. Notation as in Lemma 33.5.1. Assume $X$ is locally of finite type over $k$. Then

\[ \dim (\mathcal{O}_{Y, y}/\mathfrak m_ x\mathcal{O}_{Y, y}) = \text{trdeg}_ k(\kappa (x)) - \text{trdeg}_ K(\kappa (y)) = \dim (\mathcal{O}_{Y, y}) - \dim (\mathcal{O}_{X, x}) \]

Proof. This is a restatement of Algebra, Lemma 10.116.7. $\square$

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