Gorenstein ring

A commutative local Noetherian ring of finite injective dimension (cf. Homological dimension). A local ring with a maximal ideal and residue field of dimension is a Gorenstein ring if and only if one of the following equivalent conditions is satisfied:

1) for and .

2) For any maximal -sequence (cf. Depth of a module) the ideal is irreducible.

3) The functor , defined on the category of -modules of finite length, is isomorphic to the functor , where is the injective envelope of .

4) The ring is a Cohen–Macaulay ring (in particular, all local cohomology groups for ), and coincides with the injective envelope of .

5) For any -module of finite type there exists a canonical isomorphism

(local duality).

Examples of Gorenstein rings include regular rings and also their quotient rings by an ideal generated by a regular sequence of elements (complete intersections).

If a Gorenstein ring is a one-dimensional integral domain, then this ring has the following numerical characterization. Let be the integral closure of in its field of fractions, let be the conductor (cf. Conductor of an integral closure) of in , let , and let . The ring is then a Gorenstein ring if and only if . This equality was first demonstrated by D. Gorenstein [1] for the local ring of an irreducible plane algebraic curve. A localization of a Gorenstein ring is a Gorenstein ring. In this connection an extension of the concept of a Gorenstein ring arose: A Noetherian ring (or scheme) is said to be a Gorenstein ring (scheme) if all the localizations of this ring by prime ideals (or, correspondingly, all local rings of the scheme) are local Gorenstein rings (in the former definition).

References

[1]	D. Gorenstein, "An arithmetic theory of adjoint plane curves" Trans. Amer. Math. Soc. , 72 (1952) pp. 414–436
[2]	J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959)
[3]	L.L. Abramov, E.S. Golod, "Homology algebra of the Koszul complex of a local Gorenstein ring" Math. Notes , 9 : 1 (1971) pp. 30–32 Mat. Zametki , 9 : 1 (1971) pp. 53–58
[4]	A. Grothendieck, "Géométrie formelle et géométrie algébrique" Sem. Bourbaki , 11 (1958–1959)
[5]	R. Hartshorne, "Local cohomology, a seminar given by A. Grothendieck" , Springer (1967)
[6]	R. Hartshorne, "Residues and duality" , Springer (1966)
[7]	H. Bass, "On the ubiquity of Gorenstein rings" Math. Z. , 82 (1963) pp. 8–28