The Stacks project

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.

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

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.

  1. 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.

  2. 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 37.60.5 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

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 on

OK, thanks. I've added this [here]{}. 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

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

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AWV. Beware of the difference between the letter 'O' and the digit '0'.