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 (Hartshorne 1967)). The name comes from a duality property of singular plane curves studied by Gorenstein (1952) (who was fond of claiming that he did not understand the definition of a Gorenstein ring). The zero-dimensional case had been studied by Macaulay (1934). Serre (1961) and Bass (1963) publicized the concept of Gorenstein rings.
For Noetherian local rings, there is the following chain of inclusions.
- Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings
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 ExtiR(k, R) is zero for i ≠ n and ExtnR(k, R) is isomorphic to k;
- ExtiR(k, R) = 0 for some i > n;
- ExtiR(k, R) = 0 for all i < n and ExtnR(k, R) is isomorphic to 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.
- 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:
- The ring 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:
- The ring R is not Gorenstein because the socle has dimension 2 (not 1) as a k-vector space, spanned by x and y.
- 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 Him(M) is dual to Extn−iR(M, R) 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 f(t) = ∑j dimk(Rj) tj. 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
- 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.
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