Namespaces
Variants
Actions

Difference between revisions of "Gorenstein ring"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
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:
+
<!--
 +
g0446201.png
 +
$#A+1 = 33 n = 0
 +
$#C+1 = 33 : ~/encyclopedia/old_files/data/G044/G.0404620 Gorenstein ring
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
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" />.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
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.
+
A commutative local [[Noetherian ring|Noetherian ring]] of finite injective dimension (cf. [[Homological dimension|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:
  
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" />.
+
1) $  \mathop{\rm Ext} _ {A}  ^ {i} ( k, A) = 0 $
 +
for  $  i \neq n $
 +
and  $  \mathop{\rm Ext} _ {A}  ^ {n} ( k, A) \simeq k $.
  
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" />.
+
2) For any maximal  $  A $-
 +
sequence  $  x _ {1} \dots x _ {n} $(
 +
cf. [[Depth of a module|Depth of a module]]) the ideal  $  ( x _ {1} \dots x _ {n} ) $
 +
is irreducible.
  
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
+
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 $.
  
<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>
+
4) The ring  $  A $
 +
is a [[Cohen–Macaulay ring|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).
 
(local duality).
Line 17: Line 51:
 
Examples of Gorenstein rings include regular rings and also their quotient rings by an ideal generated by a regular sequence of elements (complete intersections).
 
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).
+
If a Gorenstein ring $  A $
 +
is a one-dimensional [[Integral domain|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|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 [[#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====
 
====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>
 
<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>

Latest revision as of 19:42, 5 June 2020


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=18502
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