Gorenstein ring
In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R 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 Noetherian local 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 below. 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 sequence a1,...,an in the maximal ideal of R such that the quotient ring R/ is Gorenstein of dimension zero.
For example, if R is a commutative graded algebra over a field k 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 → R''m 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 dimension n, the following are equivalent:
- R has finite injective dimension as an R-module;
- R has injective dimension n as an R-module;
- The Ext group for i ≠ n while
- for some i > n;
- for all i < n and
- R is an n-dimensional Gorenstein ring.
Examples
- Every local complete intersection ring, in particular every regular local ring, is Gorenstein.
- 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.
- The canonical module of a Gorenstein local ring R is isomorphic to R. In geometric terms, it follows that the standard dualizing complex of a Gorenstein scheme X over a field is simply a line bundle ; this line bundle is called the canonical bundle of X. Using the canonical bundle, Serre duality takes the same form for Gorenstein schemes as in the smooth case.
- For a Gorenstein local ring of dimension n, Grothendieck local duality takes the following form. Let E be the injective hull of the residue field k as an R-module. Then, for any finitely generated R-module M and integer i, the local cohomology group is dual to in the sense that:
- Stanley showed that for a finitely generated commutative graded algebra R 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. In 2011, Miles Reid extended this structure theorem to case of codimension 4.