The Stacks project

49.16 Quasi-finite Gorenstein morphisms

This section discusses quasi-finite Gorenstein morphisms.

Lemma 49.16.1. Let $f : Y \to X$ be a quasi-finite morphism of Noetherian schemes. The following are equivalent

  1. $f$ is Gorenstein,

  2. $f$ is flat and the fibres of $f$ are Gorenstein,

  3. $f$ is flat and $\omega _{Y/X}$ is invertible (Remark 49.2.11),

  4. for every $y \in Y$ there are affine opens $y \in V = \mathop{\mathrm{Spec}}(B) \subset Y$, $U = \mathop{\mathrm{Spec}}(A) \subset X$ with $f(V) \subset U$ such that $A \to B$ is flat and $\omega _{B/A}$ is an invertible $B$-module.

Proof. Parts (1) and (2) are equivalent by definition. Parts (3) and (4) are equivalent by the construction of $\omega _{Y/X}$ in Remark 49.2.11. Thus we have to show that (1)-(2) is equivalent to (3)-(4).

First proof. Working affine locally we can assume $f$ is a separated morphism and apply Lemma 49.15.1 to see that $\omega _{Y/X}$ is the zeroth cohomology sheaf of $f^!\mathcal{O}_ X$. Under both assumptions $f$ is flat and quasi-finite, hence $f^!\mathcal{O}_ X$ is isomorphic to $\omega _{Y/X}[0]$, see Duality for Schemes, Lemma 48.21.6. Hence the equivalence follows from Duality for Schemes, Lemma 48.25.10.

Second proof. By Lemma 49.10.2, we see that it suffices to prove the equivalence of (2) and (3) when $X$ is the spectrum of a field $k$. Then $Y = \mathop{\mathrm{Spec}}(B)$ where $B$ is a finite $k$-algebra. In this case $\omega _{B/A} = \omega _{B/k} = \mathop{\mathrm{Hom}}\nolimits _ k(B, k)$ placed in degree $0$ is a dualizing complex for $B$, see Dualizing Complexes, Lemma 47.15.8. Thus the equivalence follows from Dualizing Complexes, Lemma 47.21.4. $\square$

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 ├ętale 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.

Remark 49.16.3. Let $S$ be a Noetherian scheme endowed with a dualizing complex $\omega _ S^\bullet $. Let $f : Y \to X$ be a quasi-finite Gorenstein morphism of compactifyable schemes over $S$. Assume moreover $Y$ and $X$ Cohen-Macaulay and $f$ ├ętale at the generic points of $Y$. Then we can combine Duality for Schemes, Remark 48.23.4 and Remark 49.16.2 to see that we have a canonical isomorphism

\[ \omega _ Y = f^*\omega _ X \otimes _{\mathcal{O}_ Y} \omega _{Y/X} = f^*\omega _ X \otimes _{\mathcal{O}_ Y} \mathcal{O}_ Y(R) \]

of $\mathcal{O}_ Y$-modules. If further $f$ is finite, then the isomorphism $\mathcal{O}_ Y(R) = \omega _{Y/X}$ comes from the global section $\tau _{Y/X} \in H^0(Y, \omega _{Y/X})$ which corresponds via duality to the map $\text{Trace}_ f : f_*\mathcal{O}_ Y \to \mathcal{O}_ X$, see Lemma 49.15.2.

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