The Stacks project

Lemma 49.9.2. Let $f : Y \to X$ be a flat quasi-finite morphism of Noetherian schemes. Let $V = \mathop{\mathrm{Spec}}(B) \subset Y$, $U = \mathop{\mathrm{Spec}}(A) \subset X$ be affine open subschemes with $f(V) \subset U$. If the Dedekind different of $A \to B$ is defined, then

\[ \mathfrak {D}_ f|_ V = \widetilde{\mathfrak {D}_{B/A}} \]

as coherent ideal sheaves on $V$.

Proof. This is clear from Lemmas 49.8.1 and 49.8.3. $\square$

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