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 $x \in X$.
48.24 Gorenstein schemes
This section is the continuation of Dualizing Complexes, Section 47.21.
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
$X$ is Gorenstein $\Leftrightarrow $ $\omega _ X^\bullet $ is an invertible object of $D(\mathcal{O}_ X)$,
$\mathcal{O}_{X, x}$ is Gorenstein $\Leftrightarrow $ $\omega _{X, x}^\bullet $ is an invertible object of $D(\mathcal{O}_{X, x})$,
$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. $\square$
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