In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, 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
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
''m'', the ''R''/''m''
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
vanishes if ''n'' ≠ height(''m'') and is
1-dimensional if ''n'' = height(''m'').
A dualizing module need not be unique because the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
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 In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring ''R'' with finite injective dimension as an ''R''-module. There are many equivalent conditions, some of them listed below, often saying that 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''(''x''
2,''y''
2,''xy'') has a unique dualizing module, but it is not isomorphic to ''R''.
The ring Z[] has two non-isomorphic dualizing modules, corresponding to the two classes of invertible ideals.
The local ring ''k''
'x'',''y''(''y''
2,''xy'') is not Cohen–Macaulay so does not have a dualizing module.
See also
*
dualizing sheaf
In algebraic geometry, the dualizing sheaf on a proper scheme ''X'' of dimension ''n'' over a field ''k'' is a coherent sheaf \omega_X together with a linear functional
:t_X: \operatorname^n(X, \omega_X) \to k
that induces a natural isomorphism of ...
References
*
*{{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
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambr ...
, series=Cambridge Studies in Advanced Mathematics , isbn=978-0-521-41068-7 , mr=1251956 , year=1993 , volume=39
Commutative algebra