# 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  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).

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