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
How to Cite This Entry:
Gorenstein ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gorenstein_ring&oldid=47105
This article was adapted from an original article by V.I. DanilovI.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article