View source for Gorenstein ring

A commutative local [[Noetherian ring|Noetherian ring]] of finite injective dimension (cf. [[Homological dimension|Homological dimension]]). A local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g0446201.png" /> with a maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g0446202.png" /> and residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g0446203.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g0446204.png" /> is a Gorenstein ring if and only if one of the following equivalent conditions is satisfied:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g0446205.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g0446206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g0446207.png" />.

2) For any maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g0446208.png" />-sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g0446209.png" /> (cf. [[Depth of a module|Depth of a module]]) the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462010.png" /> is irreducible.

3) The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462011.png" />, defined on the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462012.png" />-modules of finite length, is isomorphic to the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462014.png" /> is the injective envelope of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462015.png" />.

4) The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462016.png" /> is a [[Cohen–Macaulay ring|Cohen–Macaulay ring]] (in particular, all local cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462017.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462018.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462019.png" /> coincides with the injective envelope of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462020.png" />.

5) For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462021.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462022.png" /> of finite type there exists a canonical isomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462023.png" /></td> </tr></table>

(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462024.png" /> is a one-dimensional [[Integral domain|integral domain]], then this ring has the following numerical characterization. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462025.png" /> be the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462026.png" /> in its field of fractions, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462027.png" /> be the conductor (cf. [[Conductor of an integral closure|Conductor of an integral closure]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462029.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462030.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462031.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462032.png" /> is then a Gorenstein ring if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044620/g04462033.png" />. This equality was first demonstrated by D. Gorenstein [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gorenstein,   "An arithmetic theory of adjoint plane curves"  ''Trans. Amer. Math. Soc.'' , '''72'''  (1952)  pp. 414–436</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,   "Groupes algébrique et corps des classes" , Hermann  (1959)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Grothendieck,   "Géométrie formelle et géométrie algébrique"  ''Sem. Bourbaki'' , '''11'''  (1958–1959)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R. Hartshorne,   "Local cohomology, a seminar given by A. Grothendieck" , Springer  (1967)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Hartshorne,   "Residues and duality" , Springer  (1966)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  H. Bass,   "On the ubiquity of Gorenstein rings"  ''Math. Z.'' , '''82'''  (1963)  pp. 8–28</TD></TR></table>