Gorenstein ring

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 (equivalently, with R of finite length as an R-module) is Gorenstein if and only if HomR(k, R) 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,...,a**n in the maximal ideal of R such that the quotient ring R/( a1,...,a**n) 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 ⊕ ... ⊕ R**m, then R is Gorenstein if and only if it satisfies Poincaré duality, meaning that the top graded piece R**m has dimension 1 and the product R**a × R**m−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 (hence of dimension zero as a ring) is Gorenstein if and only if there is an F-linear map e: R → F such that the symmetric bilinear form (x, y) := e(xy) on R (as an F-vector space) is nondegenerate.

For a commutative Noetherian 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;
• The Ext group \operatorname{Ext}^i_R(k,R) = 0 for i ≠ n while \operatorname{Ext}^n_R(k,R) \cong k;
• \operatorname{Ext}^i_R(k,R) = 0 for some i n;
• \operatorname{Ext}^i_R(k,R) = 0 for all i \operatorname{Ext}^n_R(k,R) \cong 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, R is said to be a local Gorenstein ring.

Examples

• Every local complete intersection ring, in particular every regular local ring, is Gorenstein.

• The ring R = k[x,y,z]/(x2, y2, xz, yz, z2−xy) 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: {1,x,y,z,z^2}. 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 (not 3) relations.

• The ring R = k[x,y]/(x2, y2, xy) 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: {1,x,y}. R is not Gorenstein because the socle has dimension 2 (not 1) 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 (viewed as a complex in degree −dim(X)); 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.

:In the context of graded rings R, the canonical module of a Gorenstein ring R is isomorphic to R with some degree shift.

• For a Gorenstein local ring (R, m, k) of dimension n, Grothendieck local duality takes the following form. Let E(k) 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 H^i_m(M) is dual to \operatorname{Ext}_R^{n-i}(M,R) in the sense that: