Gorenstein Rings and Duality

Gorenstein rings are a class of Cohen-Macaulay rings with a self-duality property analogous to Poincare duality for manifolds. They play a central role in duality theory, algebraic geometry, and homological algebra.

Definition

A Noetherian local ring $(R, \mathfrak{m}, k)$ is Gorenstein if it has finite injective dimension as a module over itself: $\operatorname{id}_R(R) < \infty$ . Equivalently, $R$ is Gorenstein if it is Cohen-Macaulay and its type $r(R) = \dim_k \operatorname{Ext}^d_R(k, R)$ equals $1$ , where $d = \dim R$ .

The hierarchy of ring classes is: $\text{regular} \implies \text{complete intersection} \implies \text{Gorenstein} \implies \text{Cohen-Macaulay}$

Definition

The canonical module of a Cohen-Macaulay local ring $(R, \mathfrak{m})$ of dimension $d$ is a finitely generated $R$ -module $\omega_R$ such that $\omega_R \cong \operatorname{Ext}^{n-d}_S(R, S)$ for any surjection $S \twoheadrightarrow R$ from a regular local ring $S$ of dimension $n$ . The ring $R$ is Gorenstein if and only if $\omega_R \cong R$ .

Local Duality

Theorem8.5Local Duality

Let $(R, \mathfrak{m}, k)$ be a complete Cohen-Macaulay local ring of dimension $d$ with canonical module $\omega_R$ . For any finitely generated $R$ -module $M$ and $0 \leq i \leq d$ , there is a natural isomorphism $H^i_\mathfrak{m}(M) \cong \operatorname{Hom}_R(\operatorname{Ext}^{d-i}_R(M, \omega_R), E)^\vee$ where $H^i_\mathfrak{m}$ denotes local cohomology and $E = E_R(k)$ is the injective hull of the residue field.

ExampleGorenstein vs. non-Gorenstein

$k[[x, y]]/(x^2, xy, y^2)$ is Artinian with socle $\operatorname{Soc}(R) = \mathfrak{m} = (\bar{x}, \bar{y})$ , which has $\dim_k = 2$ . Since $r(R) = 2 \neq 1$ , this ring is not Gorenstein.
$k[[x, y]]/(x^2, y^2)$ has $\operatorname{Soc}(R) = (\bar{x}\bar{y})$ , which is $1$ -dimensional. Thus $r(R) = 1$ and $R$ is Gorenstein.

RemarkDuality in algebraic geometry

For a projective variety $X$ over a field, the canonical module globalizes to the dualizing sheaf $\omega_X$ , and the duality becomes Serre duality: $H^i(X, \mathcal{F}) \cong H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X)^\vee$ . The Gorenstein condition ( $\omega_X \cong \mathcal{O}_X$ ) characterizes Calabi-Yau varieties, which are of central importance in string theory and mirror symmetry.