Section 48.24 (0AWV): Gorenstein schemes

48.24 Gorenstein schemes

This section is the continuation of Dualizing Complexes, Section 47.21.

Definition 48.24.1. Let $X$ be a scheme. We say $X$ is Gorenstein if $X$ is locally Noetherian and $\mathcal{O}_{X, x}$ is Gorenstein for all $x \in X$ .

This definition makes sense because a Noetherian ring is said to be Gorenstein if and only if all of its local rings are Gorenstein, see Dualizing Complexes, Definition 47.21.1.

Lemma 48.24.2. A Gorenstein scheme is Cohen-Macaulay.

Proof. Looking affine locally this follows from the corresponding result in algebra, namely Dualizing Complexes, Lemma 47.21.2. $\square$

Lemma 48.24.3. A regular scheme is Gorenstein.

Proof. Looking affine locally this follows from the corresponding result in algebra, namely Dualizing Complexes, Lemma 47.21.3. $\square$

Lemma 48.24.4. Let $X$ be a locally Noetherian scheme.

If $X$ has a dualizing complex $\omega _ X^\bullet$ , then
1. $X$ is Gorenstein $\Leftrightarrow$ $\omega _ X^\bullet$ is an invertible object of $D(\mathcal{O}_ X)$ ,
2. $\mathcal{O}_{X, x}$ is Gorenstein $\Leftrightarrow$ $\omega _{X, x}^\bullet$ is an invertible object of $D(\mathcal{O}_{X, x})$ ,
3. $U = \{ x \in X \mid \mathcal{O}_{X, x}\text{ is Gorenstein}\}$ is an open Gorenstein subscheme.
If $X$ is Gorenstein, then $X$ has a dualizing complex if and only if $\mathcal{O}_ X[0]$ is a dualizing complex.

Proof. Looking affine locally this follows from the corresponding result in algebra, namely Dualizing Complexes, Lemma 47.21.4. $\square$

Lemma 48.24.5. If $f : Y \to X$ is a local complete intersection morphism with $X$ a Gorenstein scheme, then $Y$ is Gorenstein.

Proof. By More on Morphisms, Lemma 37.62.5 it suffices to prove the corresponding statement about ring maps. This is Dualizing Complexes, Lemma 47.21.7. $\square$

Lemma 48.24.6. The property $\mathcal{P}(S) =$ “ $S$ is Gorenstein” is local in the syntomic topology.

Proof. Let $\{ S_ i \to S\}$ be a syntomic covering. The scheme $S$ is locally Noetherian if and only if each $S_ i$ is Noetherian, see Descent, Lemma 35.16.1. Thus we may now assume $S$ and $S_ i$ are locally Noetherian. If $S$ is Gorenstein, then each $S_ i$ is Gorenstein by Lemma 48.24.5. Conversely, if each $S_ i$ is Gorenstein, then for each point $s \in S$ we can pick $i$ and $t \in S_ i$ mapping to $s$ . Then $\mathcal{O}_{S, s} \to \mathcal{O}_{S_ i, t}$ is a flat local ring homomorphism with $\mathcal{O}_{S_ i, t}$ Gorenstein. Hence $\mathcal{O}_{S, s}$ is Gorenstein by Dualizing Complexes, Lemma 47.21.8. $\square$

Comments (4)

Comment #1314 by Liran Shaul on February 14, 2015 at 16:16

Re "Nonetheless many rings in algebraic geometry have dualizing complexes simply because they are quotients of Gorenstein rings, which are defined as follows.".

It seems natural to mention here that this condition is in fact both necessary and sufficient. That is: a noetherian ring has dualizing complexes if and only if it is a quotient of a finite dimensional Gorenstein ring. This was proved in Corollary 1.4 of

Kawasaki, T. (2002). On arithmetic Macaulayfication of Noetherian rings. Transactions of the American Mathematical Society, 354(1), 123-149

Comment #1316 by Johan on February 14, 2015 at 21:50

OK, thanks. I've added this [here]{https://github.com/stacks/stacks-project/commit/38254971765fad7f779e6f0453aa986274828f5b}. Of course, it would be even better if somebody had a, already written (?), write up of the history of dualizing complexes and their existence, which we could then add as a section to this chapter.

The initial goal for this chapter was to write just enough so it can be used for: (a) the relative dualizing sheaf of a family of curves used in dealing with moduli of curves, (b) the proof of resolution of surfaces, and (c) the finiteness theorem in local cohomology.

Comment #2256 by David Hansen on October 18, 2016 at 17:46

First sentence: "we seen" --> "we've seen".

Comment #2290 by Johan on November 03, 2016 at 23:15

Thanks. Fixed here.

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The Stacks project

48.24 Gorenstein schemes

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