Gorenstein ring
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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.
For a Noetherian commutative local ring (R,I,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;
for 
and 
is isomorphic to k;- for some i > n;
- for some i > n and 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.
[edit] Examples
- Every complete intersection ring is Gorenstein.
- Every regular local ring is a complete intersection ring, so is Gorenstein.
See also Daniel Gorenstein.
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.
[edit] References
- Hideyuki Matsumura, Commutative Ring Theory, Cambridge studies in advanced mathematics 8.
