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 $f: M \to N$ is called a module homomorphism or a R-linear map if for any x, y in M and r in R,

f(x + y) = f(x) + f(y),

f(rx) = rf(x).

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

f(xr) = f(x)r.

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 Hom_R(M, N). It is an abelian group but is not necessarily a module unless R is commutative.

The isomorphism theorems hold for module homomorphisms.

Examples

$\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}/n, \mathbb{Z}/m) = \mathbb{Z}/\operatorname{gcd}(n,m)$ .
For any ring R,
- $\operatorname{End}_R(R) = R$ as rings when R is viewed as a right module over itself.
- $\operatorname{Hom}_R(R, M) = M$ through $f \mapsto f(1)$ for any left module M.^[1]
- $\operatorname{Hom}_R(M, R)$ 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 $M^*$ .

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 $F \to M$ 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 $M \to N$ is to give a module homomorphism $F \to N$ that kills K (i.e., maps K to zero).

Operations

If $f: M \to N$ and $g: M' \to N'$ are module homomorphisms, then their direct sum is

f \oplus g: M \oplus M' \to N \oplus N', \, (x, y) \mapsto (f(x), g(y))

and their tensor product is

f \otimes g: M \otimes M' \to N \otimes N', \, x \otimes y \mapsto f(x) \otimes g(y).

Let $f: M \to N$ be a module homomorphism between left modules. The transpose of f is

f^*: N^* \to M^*, \, f^*(\alpha) = \alpha \circ f.

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

0 \to A \overset{f}\to B \overset{g}\to C \to 0

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:

0 \to A_{\mathfrak{m}} \overset{f}\to B_{\mathfrak{m}} \overset{g}\to C_{\mathfrak{m}} \to 0

where the subscript ${\mathfrak{m}}$ means the localization of a module at ${\mathfrak{m}}$ .

Any module homomorphism f fits into

0 \to K \to M \overset{f}\to N \to C \to 0

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

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

0 \to M \times_{B} N \to M \times N \overset{\phi}\to B \to 0

where $\phi(x, y) = f(x) - g(x)$ .

Example: Let $B \subset A$ 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 $A \to A/I, B/I \to A/I$ form a fiber square with $B = A \times_{A/I} B/I.$

Endomorphisms of finitely generated modules

Let $\phi: M \to M$ be an endomorphism between finitely generated R-modules for a commutative ring R. Then

$\phi$ is killed by its characteristic polynomial relative to the generators of M; see Nakayama's lemma#Proof.
If $\phi$ 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 $M \to N$ from a module M to a module N is a submodule of $M \oplus N.$ ^[3] In other words, it is a "many-valued" homomorphism defined on some submodule of M. The inverse $f^{-1}$ of f is the submodule $\{ (y, x) | (x, y) \in f \}$ . Any additive relation f determines a homomorphism from a submodule of M to a quotient of N

D(f) \to N/\{ y | (0, y) \in f \}

where $D(f)$ 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.

Notes

↑ Bourbaki, § 1.14
↑ Matsumura, Theorem 2.4.
↑

References

Bourbaki, Algera
S. MacLane, Homology
H. Matsumura, Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.

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