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.

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.

Examples

1. Every local complete intersection ring is Gorenstein.
2. Every regular local ring is a complete intersection ring, so is Gorenstein.
3. Every field is a regular local ring, so is Gorenstein.

See also

• Daniel Gorenstein

References

• Hideyuki Matsumura, Commutative Ring Theory, Cambridge studies in advanced mathematics 8.