The Stacks project

Lemma 26.18.6. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ with image $s \in S$. Thinking of $x$ as a point of $X_ s$ (see Lemma 26.18.5) we have

\[ \mathcal{O}_{X_ s, x} \cong \mathcal{O}_{X, x}/\mathfrak m_ s\mathcal{O}_{X, x} \cong \mathcal{O}_{X, x} \otimes _{\mathcal{O}_{S, s}} \kappa (s) \]

Proof. As in the proof of Lemma 26.18.5 this reduces to the case where $X$ and $S$ are affine. In the affine case the statement translates to Algebra, Remark 10.18.7. $\square$

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