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 (published in ). The name comes from a duality property of singular plane curves studied by (who was fond of claiming that he did not understand the definition of a Gorenstein ring ). 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 (equivalently, with R of finite length as an R-module) is Gorenstein if and only if Hom_R(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 <math>\operatorname{Ext}^i_R(k,R) = 0</math> for i ≠ n while <math>\operatorname{Ext}^n_R(k,R) \cong k;</math>
• <math>\operatorname{Ext}^i_R(k,R) = 0</math> for some i > n;
• <math>\operatorname{Ext}^i_R(k,R) = 0</math> for all i < n and <math>\operatorname{Ext}^n_R(k,R) \cong k;</math>
• 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: <math>\{1,x,y,z,z^2\}.</math> 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: <math>\{1,x,y\}.</math> 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 <math>H^i_m(M)</math> is dual to <math>\operatorname{Ext}_R^{n-i}(M,R)</math> in the sense that:

::<math>H_m^i(M) \cong \operatorname{Hom}_R(\operatorname{Ext}_R^{n-i}(M,R), E(k)).</math>

• 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

::<math>f(t) = \sum\nolimits_j \dim_k(R_j)t^j.</math>

:Namely, a graded domain R is Gorenstein if and only if it is Cohen–Macaulay and the Hilbert series is symmetric in the sense that

::<math>f\left(\tfrac{1}{t} \right)=(-1)^n t^s f(t)</math>

:for some integer s, where n is the dimension of R.

• Let (R, m, k) be a Noetherian local ring of embedding codimension c, meaning that c = dim_k(m/m²) − dim(R). 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.

Notes

References