# 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 (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.

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 above. 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.[1] More generally, a Noetherian local ring *R* is Gorenstein if and only if there is a regular sequence *a*_{1},...,*a*_{n} in the maximal ideal of *R* such that the quotient ring *R*/( *a*_{1},...,*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* ⊕ *R*_{1} ⊕ ... ⊕ *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*.[2]

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.[3]

For a commutative Noetherian local ring (*R*, *m*, *k*) of Krull dimension *n*, the following are equivalent:[4]

*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.

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*]/(*x*^{2},*y*^{2},*xz*,*yz*,*z*^{2}−*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:*R*is Gorenstein because the socle has dimension 1 as a*k*-vector space, spanned by*z*^{2}. 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*]/(*x*^{2},*y*^{2},*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:*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.[5]

- 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.[6]

- For a Gorenstein local ring (
*R*,*m*,*k*) of dimension*n*, Grothendieck local duality takes the following form.[7] 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 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

- 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*.[8]

- Let (
*R*,*m*,*k*) be a Noetherian local ring of embedding codimension*c*, meaning that*c*= dim_{k}(*m*/*m*^{2}) − 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.[9] 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.[10]

## Notes

- Eisenbud (1995), Proposition 21.5.
- Huneke (1999), Theorem 9.1.
- Lam (1999), Theorems 3.15 and 16.23.
- Matsumura (1989), Theorem 18.1.
- Matsumura (1989), Theorem 18.3.
- Eisenbud (1995), section 21.11.
- Bruns & Herzog (1993), Theorem 3.5.8.
- Stanley (1978), Theorem 4.4.
- Eisenbud (1995), Corollary 21.20.
- Bruns & Herzog (1993), Theorem 3.4.1.

## References

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