Relationship Between Hom and Limits (Modules)(Pt. I)
Point of Post: In this post we discuss how the Hom functor relates to limits. This shall serve, later on, to be a prime example of adjoint functors.
Now that we have properly defined notions of direct and inverse limits of modules there are several natural questions we can ask, relating to previous topics. Perhaps one of the natural questions one might ask is how direct and inverse limits react with the Hom module (group). We should already have a good idea what’s going to happen based on the case when we are taking the direct and inverse limit over trivial directed and inverse systems. In other words, we already know what happens in the case of products and coproducts. This suggests that perhaps something of the form and . Of course, we have to define what precisely what we mean here for, obviously as it stands, this makes absolutely no sense–what are the systems we are taking the limits over on the right hand side of both these isomorphisms? Regardless, the idea of why this “should” be true is clear enough. Namely direct limits are constructed to be such that mappings out them are completely determined by a set of mappings out of each of the individual terms of the limit, and similarly inverse limits are such that mappings into them are dtermined by a set of mappings into each of the individual factors. This roughly tells us that where denotes “related by some operation”, and similarly for the other entry with inverse limit. What suggests that in both cases we would get inverse limits is that there are natural projections maps and which heavily suggests we should be doing some kind of inverse limit (whenever one see mappings into things, inverse limits should be an immediate thought). Perhaps a bigger hint, or those who are in the know (or have done things in a slightly different order than I am posting), is that we know the maps and (where ) in the and case respectively naturally induce maps and respectively since, as we’ve already discussed, and are contravariant and covariant functors respectively. This heavily suggests that we are going to be doing some kind inverse limit. Making all of this rigorous is slightly annoying, but this the basic idea. Either way, the proof shouldn’t seem to foreign since we have already tackled the special case of products and coproducts.
Relationship Beween Hom and Limits
Now that we have discussed what we plan on doing, let’s get right to it. We’ll start with the identity . As was previously stated we have to first define what inverse system we are taking a limit over–to this end we suppose we have already been given our preordered set and our direct system and assume that is our direct limit (note that we are assuming a direct limit of any form, not necessarily the standard, this makes things more clear). The key hint was (as was noted in the motivation) the fact that is naturally a contravariant functor. Indeed, each map naturally induces a map given by
What we claim then is that is an inverse system. The one small point I’ve glossed over is: an inverse system of what? Namely, we know that the Hom groups are, in general, just abelian groups (at least naturally) and aren’t -modules unless we assume that is commutative. Of course, this distinction is unimportant since what we have created is (always) an inverse system of abelian groups (i.e. -modules) and when permitting (i.e. when is commutative) an inverse system of -modules. Moreover, the projection maps will (always) be group homomorphisms, and when permitting will be -maps. That said, while the distinction (insofar as the proofs are concerned) is artificial, it’s something important to keep in mind. Anyways, to verify that the system we have in mind is, in fact, an inverse system we must really only check that and whenenver .The first of these is trivial since for any one has that , and so is identity as desired. Thus, it remains to show that composition works out the way we want it to. But, this is just a simple computation, namely given any we have that
from where the desired conclusion follows. So, now that we know that this is a direct system, what we claim is that is an inverse limit of this system, with the projections given by . Indeed:
Theorem: Let be preordered set and be any direct system of -modules with (some) direct limit . Then, is an inverse limit of the inverse system (as described above) where this isomorphism is of abelian groups, and moreover of -modules if is commutative.
Proof: Throughout the proof we let be the ring or , so that we can do both cases at once. We begin by noting that the “projection” maps are -maps that satisfy the desired equations since
for all and . Suppose now that we are given an -module and a set of -maps such that . We then note that if we fix the set is a set of maps such that . Thus, by the definition of the direct limit we are guaranteed a map such that . Let by defined by . We next note that is an -map since
and, as we can recall, maps are completely determined by their precomposition with each . Next, note that (by definition of ) for each so that for each . Thus, is the desired map. Moreover, it’s clear that such a map is unique, for if is another -map such that then we see for each that and so the agree on each and so, by the definition of the direct limit, must agree on . Thus, , and since was arbitrary we may conclude that . Thus, we see that each set of -maps into the lifts uniquely to a map into which is compatible with the ‘s. From this we may conclude that is an inverse limit as desired.
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