The Stacks project

Remark 49.16.2. Let $f : Y \to X$ be a quasi-finite Gorenstein morphism of Noetherian schemes. Let $\mathfrak D_ f \subset \mathcal{O}_ Y$ be the different and let $R \subset Y$ be the closed subscheme cut out by $\mathfrak D_ f$. Then we have

1. $\mathfrak D_ f$ is a locally principal ideal,

2. $R$ is a locally principal closed subscheme,

3. $\mathfrak D_ f$ is affine locally the same as the Noether different,

4. formation of $R$ commutes with base change,

5. if $f$ is finite, then the norm of $R$ is the discriminant of $f$, and

6. if $f$ is étale in the associated points of $Y$, then $R$ is an effective Cartier divisor and $\omega _{Y/X} = \mathcal{O}_ Y(R)$.

This follows from Lemmas 49.9.3, 49.9.4, and 49.9.7.