In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ringR with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense. Gorenstein rings were introduced by Grothendieck in his 1961 seminar. The name comes from a duality property of singular plane curves studied by . The zero-dimensional case had been studied by. and publicized the concept of Gorenstein rings. Frobenius rings are noncommutative analogs of zero-dimensional Gorenstein rings. Gorenstein schemes are the geometric version of Gorenstein rings. For Noetherianlocal rings, there is the following chain of inclusions.
Definitions
A Gorenstein ring is a commutative Noetherian ring such that each localization at a prime ideal is a Gorenstein local ring, as defined above. A Gorenstein ring is in particular Cohen–Macaulay. One elementary characterization is: a Noetherian local ring R of dimension zero is Gorenstein if and only if HomR has dimension 1 as a k-vector space, where k is the residue field of R. Equivalently, R has simple socle as an R-module. More generally, a Noetherian local ring R is Gorenstein if and only if there is a regular sequencea1,...,an in the maximal ideal of R such that the quotient ringR/ is Gorenstein of dimension zero. For example, if R is a commutative graded algebra over a fieldk such that R has finite dimension as a k-vector space, R = k ⊕ R1 ⊕... ⊕ Rm, then R is Gorenstein if and only if it satisfies Poincaré duality, meaning that the top graded piece Rm has dimension 1 and the product Ra × Rm−a → Rm is a perfect pairing for every a. Another interpretation of the Gorenstein property as a type of duality, for not necessarily graded rings, is: for a field F, a commutative F-algebra R of finite dimension as an F-vector space is Gorenstein if and only if there is an F-linear map e: R → F such that the symmetric bilinear form := e on R is nondegenerate. For a commutative Noetherian local ring of Krull dimensionn, the following are equivalent:
A 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, R is said to be a local Gorenstein ring.
The ring R = k/ is a 0-dimensional Gorenstein ring that is not a complete intersection ring. In more detail: a basis for R as a k-vector space is given by: R is Gorenstein because the socle has dimension 1 as a k-vector space, spanned by z2. Alternatively, one can observe that R satisfies Poincaré duality when it is viewed as a graded ring with x, y, z all of the same degree. Finally. R is not a complete intersection because it has 3 generators and a minimal set of 5 relations.
The ring R = k/ is a 0-dimensional Cohen–Macaulay ring that is not a Gorenstein ring. In more detail: a basis for R as a k-vector space is given by: R is not Gorenstein because the socle has dimension 2 as a k-vector space, spanned by x and y.
Properties
A Noetherian local ring is Gorenstein if and only if its completion is Gorenstein.
Stanley showed that for a finitely generated commutative graded algebraR over a field k such that R is an integral domain, the Gorenstein property depends only on the Cohen–Macaulay property together with the Hilbert series
Let be a Noetherian local ring of embedding codimension c, meaning that c = dimk − dim. In geometric terms, this holds for a local ring of a subscheme of codimension c in a regular scheme. For c at most 2, Serre showed that R is Gorenstein if and only if it is a complete intersection. There is also a structure theorem for Gorenstein rings of codimension 3 in terms of the Pfaffians of a skew-symmetric matrix, by Buchsbaum and Eisenbud.
