Gorenstein ring

From Wikipedia, the free encyclopedia

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.

A Gorenstein commutative 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.

The classical definition reads:

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.

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.

A noteworthy occurrence of the concept is as one ingredient (among many) of the solution by Andrew Wiles to the Fermat Conjecture.