The Hom-in functor and injective modules

#module_theory

For every object $r$ in a category $C$ , we can consider the "hom-in" functor that is dual to the hom-out functor, that is the functor ${Hom}_{C} (-, r)$ . In general this is a functor from $C^{op}$ to $S e t$ , although just as with the hom-out functor in the case of $R$ -modules we can consider it a functor with values in the category of abelian groups.

We now analyze the properties of this functor, in parallel with those of the hom-out functor.

The hom-in functor and direct products, sums

How does the hom-in functor interact with direct products? At first glance, it seems somewhat differently than the hom-out functor:

The hom-out functor exchanges direct sums for direct products

Suppose $D$ is an $R$ -module and ${N_{a} ∣ a \in A}$ is a family of $R$ -modules. There is an $R$ -module isomorphism

{Hom}_{R} (⨁_{a \in A} N_{a}, D) \tilde{\to} \prod_{a \in A} {Hom}_{R} (N_{a}, D),

given by sending a morphism $f : ⨁_{a \in A} N_{a} \to D$ to the family of morphisms $(f \circ i_{a})_{a \in A}$ , where $i_{a} : N_{a} \to ⨁_{a^{'} \in A} N_{a^{'}}$ is the canonical injection.

Recall that for finite families the direct product and direct sum constructions are isomorphic, so in that case we can replace the direct product with a direct sum. Because of this, we might sometimes say that "the hom-in functor commutes with finite direct sums." The above result is a more honest picture of the general case, though.

This is actually directly analogous to the property for the hom-out functor, once we recall that the hom-in functor is a contravariant functor; i.e., ${Hom}_{R} (-, D) : (R - Mod)^{op} \to Ab$ . Note that direct products in the category $(R - Mod)^{op}$ correspond to direct sums in the category $R - Mod$ . So the above isomorphism can be viewed as the statement that the functor ${Hom}_{R} (-, D) : (R - Mod)^{op} \to Ab$ commutes with direct products in the domain and codomain categories. This is the identical property enjoyed by the functor ${Hom}_{R} (D, -) : R - Mod \to Ab$ .

The hom-in functor and exact sequences

The hom-in functor is left exact

Suppose we have an exact sequence of $R$ -modules

M \overset{g}{\to} N \to 0.

Then the corresponding sequence of abelian groups

0 \to {Hom}_{R} (N, D) \overset{- \circ g}{\to} {Hom}_{R} (M, D)

is also exact.

More generally, if we have a short exact sequence of $R$ -modules

0 \to L \overset{f}{\to} M \overset{g}{\to} N \to 0,

then the corresponding sequence of abelian groups below is also exact:

0 \to {Hom}_{R} (N, D) \overset{- \circ g}{\to} {Hom}_{R} (M, D) \overset{- \circ f}{\to} {Hom}_{R} (L, D) .

Notice that the $0$ on the far right of the sequence is gone! We have lost the "right end" of our exact sequence. Because of the above property, we say that the functor ${Hom}_{R} (-, D) : R - Mod \to Ab$ is left exact.

Is it ever the case that the functor ${Hom}_{R} (-, D)$ is exact, in other words sends short exact sequences to short exact sequences?

Definition of an injective module

An $R$ -module $Q$ is injective if for every short exact sequence of $R$ -modules

0 \to L \overset{f}{\to} M \overset{g}{\to} N \to 0

the corresponding sequence of abelian groups is also exact:

0 \to {Hom}_{R} (N, Q) \overset{- \circ g}{\to} {Hom}_{R} (M, Q) \overset{- \circ f}{\to} {Hom}_{R} (L, Q) \to 0.

Here is a nice characterization of injective modules:

Equivalent condition of injective

An $R$ -module $Q$ is injective if and only if whenever $Q$ is a submodule of an $R$ -module $M$ then $Q$ is a direct summand of $M$ .

Here is a nice characterization of injective $R$ -modules when $R$ is a PID:

Injective modules over PIDs

Suppose $R$ is a PID. An $R$ -module $Q$ is injective if and only if $r Q = Q$ for every nonzero $r \in R$ .

In particular, an abelian group is injective if and only if it's divisible.

Corollary

If $R$ is a PID, then any quotient of an injective $R$ -module is injective.

Examples of injective modules

Every vector space is injective.
The abelian group $Q$ is injective.
The quotient group $Q / Z$ is injective.
Any direct sum of injective $Z$ -modules is injective; e.g., $Q \oplus Q / Z$ is injective.

Examples of non-injective modules

The abelian group $Z$ is not injective.
Any nonzero finitely generated abelian group is not injective.