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
- The zero map M → N that maps every element to zero.
- A linear transformation between vector spaces.
- .
- For a commutative ring R and ideals I, J, there is the canonical identification
- given by . In particular, is the annihilator of I.
- Given a ring R and an element r, let denote the left multiplication by r. Then for any s, t in R,
- .
- That is, is right R-linear.
- For any ring R,
- as rings when R is viewed as a right module over itself. Explicitly, this isomorphism is given by the left regular representation .
- through for any left module M.[1] (The module structure on Hom here comes from the right R-action on R; see #Module structures on Hom below.)
- is called the dual module of M; it is a left (resp. right) module if M is a right (resp. left) module over R with the module structure coming from the R-action on R. It is denoted by .
- Given a ring homomorphism R → S of commutative rings and an S-module M, an R-linear map θ: S → M is called a derivation if for any f, g in S, θ(f g) = f θ(g) + θ(f) g.
- If S, T are unital associative algebras over a ring R, then an algebra homomorphism from S to T is a ring homomorphism that is also an R-module homomorphism.
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 M ⊕ N given by
- .
It is a module since it is the image of the graph morphism M → M ⊕ N, 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
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
- is killed by its characteristic polynomial relative to the generators of M; see Nakayama's lemma#Proof.
- If is surjective, then it is injective.[2]
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
References
- Bourbaki, Algebra
- S. MacLane, Homology
- H. Matsumura, Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.