## Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. II)

**Point of Post: **This is a continuation of this post.

Ok, fine. So we know now that serves as a functor , but how “nice” of a functor is it? Well, while we have yet to discuss the categorical machinery to precisely say how nice it is, we can nonetheless prove that it has this property (e.g. we lack the language to put the following theorem on the proper theoretical ground, but we can prove and use the theorem nonetheless). Roughly what we’d like to show is that the functor perserves exactness. Of course, we have yet to define what exactness should mean for a chain of directed systems, but this isn’t too hard. Namely, we call a chain , where , , and , *exact* if the chains of -maps are exact for each . Our meaning then is clear, we’d like to show that carries exact sequences to exact sequences. In fact, to be honest we need to assume that is, in fact, directed, but since most real-life examples of preordered sets are directed, this is really no loss. In particular:

**Theorem: ***Let be elements of listed as above, with limits cones , , and respectively, and assume is directed). Suppose further that we have an exact sequence , then is exact.*

**Proof: **We begin by showing that , to this end suppose that . Recalling our characterization of direct limits over directed sets we know that there exists some and some such that . But, we then have that . Thus, we see that , but using the same characterization of direct limits over directed sets as we previously mentioned, there exists some such that . Thus, , but by exactness we may then conclude that there exists such that . We see then that

and so . Conversely, to prove that it suffices to show that . But, by definition of it suffices to show that for all . But, this is easy since

From these two inclusions we may conclude that , and so exactness follows.

Since obviously takes the zero module to the zero module, and the zero map to the zero map we get the following corollary:

**Corollary: ***Let and be directed systems such that each is injective, then is injective. Similarly, if each is surjective, then is surjective.*

**Proof: **Since each is injective we have that each is exact, and so by the above theorem and the fact mentioned in the previous paragraph we may conclude that is also exact, and so is injective. The case for surjectivity follows similarly by noting that the exact sequences get sent to the exact sequence .

Ok, let’s get more down to earth and look at some examples of how this functor plays out, in some specific cases. Suppose we start with two sets and of -modules and we considered it as a directed system over where is, as usual, the trivial directed system. Suppose further that we have have any set of maps . We note then that trivially (by default even) is a morphism in the category since the only thing we must check is that (this is because the only comparable things in are elements to themselves). From this we conclude that we may apply the functor to get an -map . Note though that by previous discussion and are nothing more than the coproducts and respectively. What we now claim is that is the map often denoted which sends to . To see this we merely note that where are the natural inclusions and respectively. But, note that and are limit cones for and , and we know by construction that satisfies this same equation. Thus, by the definition of direct limit we may conclude that .

Thus, we see from this and the above general theory that if we have a set of exact chains that we may conclude that the chain is also exact (this of course also implies that taking direct sums preserves injectivity and surjectivity). The most common case of this situation is when we have a single exact sequence and we take the -fold coproduct of the chain to conclude that is exact.

**References:**

[1] Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. *Advanced Modern Algebra*. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. *Module Theory.* Clarendon, 1990. Print.

[4] Lang, Serge. *Algebra*. Reading, MA: Addison-Wesley Pub., 1965. Print.

[5] Grillet, Pierre A. *Abstract Algebra*. New York: Springer, 2007. Print.

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