Injective Modules (Pt. I)
Point of Post: In this post we discuss the notion of injective modules and show that the category has “enough injectives”.
We are going to discuss now the “dual” notion to projective modules which, as one would expect, are just the modules one gets by dualizing the lifting axioms for projective modules. Of course, it should also follow then that we can dualize the other properties of projective modules to get other characterizations, is exact, etc.
Our main use for injective modules, similar to the case of projective modules, is that injective modules are “nice” modules for which we can effectively approximate any other module (i.e. we can find some long exact sequence beginning with a given module and with the rest of the terms in the sequence being injective modules). This will be key for when we discuss the notion of derived functors.
Let be any ring. We call a left -module injective if whenever we have a diagram of the form
We can find an -map making the resulting diagram commute.
What this roughly translates to is that every -map from a submodule of a module into can be extended to an -map . So, for example, is not an injective -module for if one tried to extend the identity map to a group map one would be able to find such that (namely, ) but this is impossible. So, one can’t extend the map to a map and so is not an injective -module. For similar reasons is not an injective -module since the identity map cannot be extended to a map since isn’t solvable in .
Let us prove two quick things about injective modules:
Theorem: Direct summands and products of injective modules are injective.
Proof: Suppose first that is any module and there exists another module such that is injective. Suppose then that we have an injective map and a map . We note then that we also have the map (where is the usual inclusion ). Since is injective there exists such that . Define then by where is the usual projection. This is clearly an -map and satisfies . Since everything was arbitrary it follows that is injective.
The fact that the product of injectives is injective follows from the fact that the is naturally isomorphic to .
Of course, while this is the best definition in the sense that it captures what injective modules “do” it is somewhat transparent.
Perhaps one of the clearest definitions of injective modules is given by the following:
Theorem: Let be a ring. Then, a left -module is injective if and only if every short exact sequence
Proof: Let’s first assume that is injective and suppose that we have a sequence . Note that we then have the following commutative diagram
so there exists an arrow such that . Thus, by the splitting lemma, we may conclude that our sequence splits.
We will prove in just a little bit that every left -module can be embedded into an injective -module. So, suppose that satisfies the splitting property. We can find an injective module such that . But, because of ‘s splitting property this says that is a direct summand of and thus injective.
It then becomes even more obvious, for example, that is not an injective -module since is not split.
This also allows us to interpret injective modules as precisely the modules that always sit nicely inside any module containing them. More particularly:
Theorem: Let be an injective module. Then, if then is a direct summand of (i.e. there exists with ).
Proof: We know that splits and so and moreover maps isomorphically onto .
We now prove the equivalence that injective modules are just those that make the contaravariant Hom functor exact:
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