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.
Point of Post: In this post we discuss and motivate the notion of projective modules, and give/prove the standard equivalences between the most common definitions.
In this post we give the definition of a type of module which, among others, shall occupy a fair amount of our attention in the math (specifically the homological algebra) to come. There are, as there were for tensor products, a multitude of (seemingly) disparate ways to motivate the usefulness/interestingness of projective modules, some of which I understand better than others. A motivation which is somewhat high on the usefulness scale but which is, perhaps, less so on the scale of interestingness is that projective modules are precisely the answer to the following question: “we know that given a module the covariant Hom functor is left-exact, for which modules is the functor actually exact?” Another question for which projective modules answer quite nicely is the question as to which modules have the property that any short exact sequence ending in them splits (for example, we have [perhaps it’s slightly opaque from the presentation in that post] seen that this is true for free modules).
While the first of these two motivations is very important both of these motivational characterization of projective modules are still, well, not so motivating. So, let’s consider some of the other questions that these projective modules seem to answer.
One of the more down-to-earth reasons why one would want to consider projective modules is concerned with a certain ‘lifting’ property of free modules. Namely, suppose that we have modules and a free module . Then, given any module epimorphism and we can create a module homomorphism such that . For free modules it is clear how to do this–we just do it (we define the mapping how we want on a basis and just extend). Of course, this universal ability to ‘factor’ doesn’t always hold for non-free modules. For example, we shall see that if then does NOT satisfy this property in the category of abelian groups. Projective modules are precisely the modules for which we can always lift such maps.
Ok, let’s discuss yet another important question for which the class of projective modules is the answer. Suppose that we made a decision to try to understand module theory by attempting to construct all modules, going through them systematically. Being the good little math students that we are, we construct the most obvious modules first–free modules. I mean, these are just the answer to “what’s the module I can construct that requires the ‘least thought'”. Once we are done looking at free modules the next obvious thing we can look at is submodules of free modules. But, once again, being the good little math students that we are, we wish to select the submodules of free modules which are “simplest”. By this, we mean that properties of the free module can be transferred to properties about the submodule. A way in which this can happen, for example, is if the submodule is actually an isomorphic image of the free module. But, the nicest way in which THAT can happen is if the submodule is actually a direct summand of the free module (i.e. there exists another module whose coproduct with the submodule is isomorphic to the ambient free module). Thus, we have decided that after free modules we should study submodules of free modules that are direct summands. Of course, the punchline should be clear by now, this class of modules is precisely the class of projective modules.
As a penultimate usefulness I mention I somewhat throwaway fact about projective modules in relation to algebraic geometry and topology which, at this point, I don’t understand very well (but would very much like to in the future). Namely, projective modules can be thought of as an algebraic analogue of trivial vector bundles in topology. This is made more precise by the Serre-Swan theorem and by this MSE answer by the very knowledgeable Georges Elencwajg. While I don’t fully understand this correspondence (it sits comfortably in the realm of K-theory) it definitely gives a somewhat fruitful geometric intuition as to what projective modules are–they represent the simplest possible pieces that a space can decompose into. It tells us that projective modules are “locally free”.
Now, for the final, and most important motivation for this (as should now be clear) ubiquitous class of modules. Namely, in the math to come (the homological algebra) we shall be very concerned with when certain left or right exact functors are actually exact. If one thinks about it, this is actually a pretty serious condition to prove. Namely, we’d theoretically have to check that every short exact sequence goes to a short exact sequence. It would be much easier if we were able to construct some kind of gadget that allowed us to determine the right or left exactness of a given left or right (respectively) exact functor just by computing one thing with this gadget. In fact, we shall be able (in most cases) to construct a gadget and for the case of starting with a right-exact functor. The integral part of this construction shall be construction a so called “projective resolution” which uses, you guessed it, projective modules. Thus, if we wish to make this construction we better have a solid understand of these strange beasts. While this aspect of motivation scores high on interestingness and usefulness it is slightly opaque as to how the previous ‘definitions’ of projective modules relate to this motivation–how we can construct such a gadget using them, and why we should expect this to be true. This is a matter better left for later discussion!
Hopefully now, you don’t just believe that projective modules are interesting, but you believe that their discussion and use was inevitable all along.
Point of Post: In this post we prove that the Hom functor is left exact.
We have proven that the tensor functor is ‘right exact’, and moreover we have seen that this partial exactness can be a great boon in our quest to compute things. But, there are other things besides tensor products for which we would like to compute. In particular, we like to compute the homomorphism group/module if not for practical reasons then for purely philosophical ones. Thus, in this post we shall prove that the Hom functor itself has a certain kind of ‘partial exactness’, one which is dual to that of the tensor product: ‘left exactness’. In other words, the Hom functor (in either variance [i.e. the covariant or contravariant]) takes exact sequences of the form to exact sequences of the same form. There should be, at least some, lack of surprise at this fact considering the somewhat dualistic nature that the Hom and tensor functors share, via their adjointness.
Point of Post: In this post we discuss more particularly about the tensor product of free modules, and some of the consequences of this.
We have proved theorems before to the effect that the tensor product of free modules are free. In this post we’d like to solidify this by discussing precisely how, given bases for two free modules, construct a basis on their tensor product. This will enable us to give an explicit example of elements of a tensor product which are not simple tensors.
Point of Post: This is a continuation of this post.
Point of Post: In this post we show how one can use partial exactness to actually compute, explicitly, the isomorphism type of certain tensor products.
Up until this point I have committed a sort of embarrassing crime–I have yet to explicitly compute a tensor product! This is not good, because I strongly believe that if one doesn’t stop and compute some things, one can get a little lost in the abstraction. So, this post is devoted primarily to discussing a way in which we can use partial exactness to explicitly compute the isomorphism type of some actual examples. Roughly the idea is that if we can express a module in terms of an exact sequence of the form then we know that is the cokernel of the initial, right exactness tells us then that is exact and so will be the cokernel of the first map. After showing some examples where this is useful we shall describe a general technique for computing tensor products by using “free presentations” which ultimately amount to expressing a module in terms of generators and relations.