The Stacks project

Lemma 49.12.3. Let $f : Y \to X$ be a morphism of Noetherian schemes. If $f$ satisfies the equivalent conditions of Lemma 49.10.1 then the different $\mathfrak {D}_ f$ of $f$ is the Kähler different of $f$.

Proof. By Lemmas 49.9.3 and 49.10.4 the different of $f$ affine locally is the same as the Noether different. Then the lemma follows from the computation of the Noether different and the Kähler different on standard affine pieces done in Lemmas 49.7.4 and 49.12.2. $\square$

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