In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, a dualizing module, also called a canonical module, is a
module over a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
that is analogous to the
canonical bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the nth exterior power of the cotangent bundle \Omega on V.
Over the complex numbers, it is ...
of a
smooth variety In algebraic geometry, a smooth scheme over a Field (mathematics), field is a scheme (mathematics), scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no Singular poi ...
. It is used in
Grothendieck local duality.
Definition
A dualizing module for a
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
''R'' is a
finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type.
Related concepts i ...
''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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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 V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
of any dualizing module with a rank 1
projective module
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizati ...
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
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...
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
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 was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, series=Cambridge Studies in Advanced Mathematics , isbn=978-0-521-41068-7 , mr=1251956 , year=1993 , volume=39
Commutative algebra