# Gorenstein ring

In commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring R with finite injective dimension, as an R-module. There are many equivalent conditions, some of them listed below, most dealing with some sort of duality condition.

Gorenstein rings were introduced by Grothendieck, who named them because of their relation to a duality property of singular plane curves studied by Gorenstein (1952) (who was fond of claiming that he did not understand the definition of a Gorenstein ring). The zero-dimensional case had been studied by Macaulay (1934). Serre (1961) and Bass (1963) publicized the concept of Gorenstein rings.

Noncommutative analogues of 0-dimensional Gorenstein rings are called Frobenius rings.

## Definitions

A Gorenstein ring is a commutative ring such that each localization at a prime ideal is a Gorenstein local ring. The Gorenstein ring concept is a special case of the more general Cohen–Macaulay ring.

A local Cohen–Macaulay ring R is called Gorenstein if there is a maximal R-regular sequence in the maximal ideal generating an irreducible ideal.[citation needed]

For a Noetherian commutative local ring $(R, m, k)$ of Krull dimension $n$, the following are equivalent:

• $R$ has finite injective dimension as an $R$-module;
• $R$ has injective dimension $n$ as an $R$-module;
• $\operatorname{Ext}^i_R (k, R) = 0$ for $i \neq n$ and $\operatorname{Ext}^n_R (k, R)$ is isomorphic to $k$;
• $\operatorname{Ext}^i_R (k, R) = 0$ for some $i > n$;
• $\operatorname{Ext}^i_R (k, R) = 0$ for all $i < n$ and $\operatorname{Ext}^n_R (k, R)$ is isomorphic to $k$;
• $R$ is an $n$-dimensional Gorenstein ring.

A (not necessarily commutative) ring R is called Gorenstein if R has finite injective dimension both as a left R-module and as a right R-module. If R is a local ring, we say R is a local Gorenstein ring.

## Examples

• The ring k[x,y,z]/(x2, y2, xz, yz, z2xy) is a 0-dimensional Gorenstein ring that is not a complete intersection ring.
• The ring k[x,y]/(x2, y2, xy) is a 0-dimensional Cohen–Macaulay ring that is not a Gorenstein ring.

## Properties

A noetherian commutative local ring is Gorenstein if and only if its completion is Gorenstein.[1]

The canonical module of a graded Gorenstein ring R is isomorphic to R with some degree shift.