The Stacks project

Remark 49.14.2. We can generalize Definition 49.9.1. Suppose that $f : Y \to X$ is a quasi-finite morphism of Noetherian schemes with the following properties

  1. the open $V \subset Y$ where $f$ is flat contains $\text{Ass}(\mathcal{O}_ Y)$ and $f^{-1}(\text{Ass}(\mathcal{O}_ X))$,

  2. the trace element $\tau _{V/X}$ comes from a section $\tau \in \Gamma (Y, \omega _{Y/X})$.

Condition (1) implies that $V$ contains the associated points of $\omega _{Y/X}$ by Lemma 49.2.8. In particular, $\tau $ is unique if it exists (Divisors, Lemma 31.2.8). Given $\tau $ we can define the different $\mathfrak {D}_ f$ as the annihilator of $\mathop{\mathrm{Coker}}(\tau : \mathcal{O}_ Y \to \omega _{Y/X})$. This agrees with the Dedekind different in many cases (Lemma 49.14.3). However, for non-flat maps between non-normal rings, this generalization no longer measures ramification of the morphism, see Example 49.14.4.

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