Lemma 49.8.3 (0BW3)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 49: Discriminants and Differents
Section 49.8: The Dedekind different
Lemma 49.8.3 (cite)

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Lemma 49.8.3. If the Dedekind different of $A \to B$ is defined and $A \to B$ is flat, then

the canonical isomorphism $\mathcal{L}_{B/A} \to \omega _{B/A}$ sends $1 \in \mathcal{L}_{B/A}$ to the trace element $\tau _{B/A} \in \omega _{B/A}$ , and
the Dedekind different is $\mathfrak {D}_{B/A} = \{ b \in B \mid b\omega _{B/A} \subset B\tau _{B/A}\}$ .

Proof. The first assertion follows from the proof of Lemma 49.8.1 and Lemma 49.4.3. The second assertion is immediate from the first and the definitions. $\square$

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