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 $i$
eq 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 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.

Contents

Examples

See also

References

Examples

# Every local complete intersection ring is Gorenstein.
# Every regular local ring is a complete intersection ring, so is Gorenstein.

See also

★ Daniel Gorenstein

References

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