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 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. 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''.

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 $\operatorname^i_R(k,R) = 0$ for ''i'' ≠ ''n'' while $\operatorname^n_R(k,R) \cong k;$
* $\operatorname^i_R(k,R) = 0$ for some ''i'' > ''n'';
* $\operatorname^i_R(k,R) = 0$ for all ''i'' < ''n'' and $\operatorname^n_R(k,R) \cong k;$
* ''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''(''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.

*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 $\operatorname_R^(M,R)$ in the sense that:
::$H_m^i(M) \cong \operatorname_R(\operatorname_R^(M,R), E(k)).$

*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
::$f\left(\tfrac \right)=(-1)^n t^s f(t)$
: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'' = 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. 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.Bruns & Herzog (1993), Theorem 3.4.1.

Notes

References

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*{{Citation , author1-last=Stanley , author1-first=Richard P. , author1-link=Richard P. Stanley , title=Hilbert functions of graded algebras , journal=Advances in Mathematics , volume=28 , issue=1 , pages=57–83 , year=1978 , doi=10.1016/0001-8708(78)90045-2 , doi-access=free , mr=0485835

See also

* Commutative algebra
* Ring theory
* Wiles's proof of Fermat's Last Theorem

Commutative algebra