Module homomorphism

In algebra, a module homomorphism is a function between modules that preserves module structures. Explicitly, if M and N are left modules over a ring R, then a function is called a module homomorphism or an R-linear map if for any x, y in M and r in R,

If M, N are right modules, then the second condition is replaced with

The pre-image of the zero element under f is called the kernel of f. The set of all module homomorphisms from M to N is denoted by HomR(M, N). It is an abelian group but is not necessarily a module unless R is commutative.

The composition of module homorphisms is again a module homomorphism. Thus, all the (say left) modules together with all the module homomorphisms between them form the category of modules.

Terminology

A module homomorphism is called an isomorphism if it admits the inverse homomorphism. A module homomorphism is an isomorphism if and only if it is an ismormosphim between the underlying abelian groups. In other words, an inverse function of a module homomorphism, when it exists, is necessary a homomorphism.

The isomorphism theorems hold for module homomorphisms.

A module homomorphism from a module M to itself is called an endomorphism and an isomorphism from M to itself an automorphism. One writes for the set of all endomorphisms between a module M. It is not only an abelian group but is also a ring with multiplication given by function composition; it is called the endomorphism ring of M.

Schur's lemma says that a homomorphism between simple modules (a module having only two submodules) must be either zero or an isomorphism. In particular, the endomorphism ring of a simple module is a division ring.

To use the language of the category theory, an injective homomorphism is also called a monomorphism and a surjective homomorphism an epimorphism.

Examples

given by . In particular, is the annihilator of I.
That is, is right R-linear.

Module structures on Hom

In short, Hom inherits a ring action that was not used up to form Hom. More precise, let M, N be left R-modules. Suppose M has a right action of a ring S that commutes with the R-action; i.e., M is an (R, S)-module. Then

has the structure of a left S-module defined by: for s in S and x in M,

It is well-defined (i.e., is R-linear) since

Similarly, is a ring action since

.

Note: the above verification would "fail" if one used the left R-action in place of the right S-action. In this sense, Hom is often said to "use up" the R-action.

Similarly, if M is a left R-module and N is an (R, S)-module, then is a right S-module by .

A matrix representation of a module homomorphism

The relationship between matrices and linear transformations in linear algebra generalizes in a natural way to module homomorphisms. Precisely, given a right R-module U, there is the canonical isomorphism of the abelian groups

by viewing consisting of column vectors and then writing f as an m × n matrix. In particular, viewing R as a right R-module, one has

,

which turns out to be a ring isomorphism.

Note the above isomorphism is canonical: if one is given a module homomorphism between finite-rank free modules, then a choice of an ordered basis corresponds to a choice of an isomorphism . The above procedure then gives the matrix representation with respect to such choices of the bases.

To define a module homomorphism

In practice, one often defines a module homomorphism by specifying its values on a generating set of a module. More precise, let M and N be left R-modules. Suppose a subset S generates M; i.e., there is a surjection with a free module F with a basis indexed by S and kernel K (i.e., the free presentation). Then to give a module homomorphism is to give a module homomorphism that kills K (i.e., maps K to zero).

Operations

If and are module homomorphisms, then their direct sum is

and their tensor product is

Let be a module homomorphism between left modules. The graph Γf of f is the submodule of MN given by

.

It is a module since it is the image of the graph morphism MMN, x → (x, f(x)).

The transpose of f is

If f is an isomorphism, then the transpose of the inverse of f is called the contragredient of f.

Exact sequences

Main article: exact sequences

A short sequence of modules over a commutative ring

consists of modules A, B, C and homomorphisms f, g. It is exact if f is injective, the kernel of g is the image of f and g is surjective. A longer exact sequence is defined in the similar way. A sequence of modules is exact if and only if it is exact as a sequence of abelian groups. Also the sequence is exact if and only if it is exact at all the maximal ideals:

where the subscript means the localization of a module at .

Any module homomorphism f fits into

where K is the kernel of f and C is the cokernel, the quotien of N by the image of f.

If are module homomorphisms, then they are said to form a fiber square (or pullback square), denoted by M ×B N, if it fits into

where .

Example: Let be commutative rings, and let I be the annihilator of the quotient B-module A/B (which is an ideal of A). Then canonical maps form a fiber square with

Endomorphisms of finitely generated modules

Let be an endomorphism between finitely generated R-modules for a commutative ring R. Then

See also: Herbrand quotient (which can be defined for any endomorphism with some finiteness conditions.)

Variants

Additive relations

An additive relation from a module M to a module N is a submodule of [3] In other words, it is a "many-valued" homomorphism defined on some submodule of M. The inverse of f is the submodule . Any additive relation f determines a homomorphism from a submodule of M to a quotient of N

where consists of all elements x in M such that (x, y) belongs to f for some y in N.

A transgression that arises from a spectral sequence is an example of an additive relation.

See also

Notes

  1. Bourbaki, § 1.14
  2. Matsumura, Theorem 2.4.

References

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