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 ter ...

, a dualizing module, also called a canonical module, is a

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...

over a

commutative ring In mathematics, a commutative ring is a 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 properties that are not ...

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 ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...

of a

smooth variety In algebraic geometry, a smooth scheme over a field is a 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 points. A special case is the notion of a smo ...

. It is used in

Grothendieck local duality In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves. Statement Suppose that ''R'' is a Cohen–Macaulay local ring of dimension ' ...

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 in ...

''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 c ...

''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 can ...

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) 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 element of V \otime ...

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, by keeping some of the main properties of free modules. Various equivalent characteriz ...

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 is ...

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 In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of Alexander Grothe ...

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 abstract algebra, more specifically 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 varieties or manifolds, or of algebrai ...

then the

Matlis module In algebra, Matlis duality is a duality between Artinian and Noetherian modules over a complete Noetherian local ring. In the special case when the local ring has a field mapping to the residue field it is closely related to earlier work by F ...

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.

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 King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pr ...

, series=Cambridge Studies in Advanced Mathematics , isbn=978-0-521-41068-7 , mr=1251956 , year=1993 , volume=39 Commutative algebra

Definition

Examples

See also

References