Gorenstein ring

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

1) $ \mathop{\rm Ext} _ {A} ^ {i} ( k, A) = 0 $ for $ i \neq n $ and $ \mathop{\rm Ext} _ {A} ^ {n} ( k, A) \simeq k $.

2) For any maximal $ A $- sequence $ x _ {1} \dots x _ {n} $( cf. Depth of a module) the ideal $ ( x _ {1} \dots x _ {n} ) $ is irreducible.

3) The functor $ M \mapsto \mathop{\rm Ext} _ {A} ^ {n} ( M, A) $, defined on the category of $ A $- modules of finite length, is isomorphic to the functor $ M \mapsto \mathop{\rm Hom} _ {A} ( M, I) $, where $ I $ is the injective envelope of $ k $.

4) The ring $ A $ is a Cohen–Macaulay ring (in particular, all local cohomology groups $ H _ {m} ^ {i} ( A) = 0 $ for $ i \neq n $), and $ H _ {m} ^ {n} ( A) $ coincides with the injective envelope of $ k $.

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

$$ H _ {m} ^ {i} ( M) \simeq \ \mathop{\rm Hom} ( \mathop{\rm Ext} ^ {n - i } ( M, A), H _ {m} ^ {n} ( A)) $$

(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 $ A $ is a one-dimensional integral domain, then this ring has the following numerical characterization. Let $ \overline{A}\; $ be the integral closure of $ A $ in its field of fractions, let $ F $ be the conductor (cf. Conductor of an integral closure) of $ A $ in $ \overline{A}\; $, let $ C = \mathop{\rm dim} _ {k} \overline{A}\; /F $, and let $ \delta = \mathop{\rm dim} _ {k} \overline{A}\; /A $. The ring $ A $ is then a Gorenstein ring if and only if $ C = 2 \delta $. 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