Dualizing module
In abstract algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety. It is used in Grothendieck local duality.
Definition
A dualizing module for a Noetherian ring R is a finitely generated module M such that for any maximal ideal m, the R/m vector space {{nowrap|Ext{{su|b=R|p=n}}(R/m,M)}} vanishes if n ≠ height(m) and is 1-dimensional if n = height(m).
A dualizing module need not be unique because the tensor product of any dualizing module with a rank 1 projective module is also a dualizing module. However this is the only way in which the dualizing module fails to be unique: given any two dualizing modules, one is isomorphic to the tensor product of the other with a rank 1 projective module.
In particular if the ring is local the dualizing module is unique up to isomorphism.
A Noetherian ring does not necessarily have a dualizing module. Any ring with a dualizing module must be Cohen–Macaulay. Conversely if a Cohen–Macaulay ring is a quotient of a Gorenstein ring then it has a dualizing module. In particular any complete local Cohen–Macaulay ring has a dualizing module. For rings without a dualizing module it is sometimes possible to use the dualizing complex as a substitute.
Examples
If R is a Gorenstein ring, then R considered as a module over itself is a dualizing module.
If R is an Artinian local ring then the Matlis module of R (the injective hull of the residue field) is the dualizing module.
The Artinian local ring R = k[x,y]/(x2,y2,xy) has a unique dualizing module, but it is not isomorphic to R.
The ring Z[{{radic|–5}}] has two non-isomorphic dualizing modules, corresponding to the two classes of invertible ideals.
The local ring k[x,y]/(y2,xy) is not Cohen–Macaulay so does not have a dualizing module.
See also
References
- {{citation|mr=2333539|last=Bourbaki|first= N.|authorlink=Nicolas Bourbaki
|series=Éléments de mathématique|title= Algèbre commutative. Chapitre 10|language=French
|publisher= Springer-Verlag, Berlin|year=2007|isbn= 978-3-540-34394-3}}
- {{Citation | last1=Bruns | first1=Winfried | last2=Herzog | first2=Jürgen | title=Cohen-Macaulay rings | url=https://books.google.com/books?id=LF6CbQk9uScC | publisher=Cambridge University Press | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-41068-7 | mr=1251956 | year=1993 | volume=39}}