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 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,...,an in the maximal ideal of R such that the quotient ring R/( a1,...,an) 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 → 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 (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 < n and $$\operatorname{Ext}^n_R(k,R) \cong k;$$
 * 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:
 * $$H_m^i(M) \cong \operatorname{Hom}_R(\operatorname{Ext}_R^{n-i}(M,R), E(k)).$$


 * 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
 * $$f(t) = \sum\nolimits_j \dim_k(R_j)t^j.$$
 * 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
 * $$f\left(\tfrac{1}{t} \right)=(-1)^n t^s f(t)$$
 * 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 = dimk(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.