Dualizing module

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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.

Abstract algebra branch of mathematics

In algebra, which is a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring and a multiplication is defined between elements of the ring and elements of the module.

In ring theory, a branch of abstract algebra, 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 noncommutative rings where multiplication is not required to be commutative.

Contents

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 Extn
R(R/m,M) vanishes if n ≠ height(m) and is 1-dimensional if n = height(m).

In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals, which means there is no infinite ascending sequence of left ideals; that is, given any chain of left ideals,

In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated R-module also may be called a finite R-module, finite over R, or a module of finite type.

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R.

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.

In mathematics, the tensor productV ⊗ W of two vector spaces V and W is itself a vector space, endowed with the operation of bilinear composition, denoted by ⊗, from ordered pairs in the Cartesian product V × W onto V ⊗ W in a way that generalizes the outer product. The tensor product of V and W is the vector space generated by the symbols v ⊗ w, with v ∈ V and w ∈ W, in which the relations of bilinearity are imposed for the product operation ⊗, and no other relations are assumed to hold. The tensor product space is thus the "freest" such vector space, in the sense of having the fewest constraints.

In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.

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.

In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.

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 self-dual in some sense.

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.

In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition.

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 algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.

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[√–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

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References

Nicolas Bourbaki collective pseudonym for a group of (mainly French) 20th-century mathematician

Nicolas Bourbaki is the collective pseudonym of a group of mathematicians. Their aim is to reformulate mathematics on an extremely abstract and formal but self-contained basis in a series of books beginning in 1935. With the goal of grounding all of mathematics on set theory, the group strives for rigour and generality. Their work led to the discovery of several concepts and terminologies still used, and influenced modern branches of mathematics.

International Standard Book Number Unique numeric book identifier

The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

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