## Inverse Limits of Modules (Pt. I)

**Point of Post: **In this post we introduce the notion of inverse limits, show uniqueness, and give some examples.

*Motivation*

As often happens when defining universal objects (i.e. those which can be defined (up to isomorphism) by some mapping properties) the notion of direct limits have a “dual notion”.Very explicitly one can define inverse limits as being the construction one gets by taking the universal characterization of direct limits and reversing all the arrows. We have already seen this kind of duality between products and coproducts, and in fact this shall serve as the main kind of duality between direct and inverse limits since intuitively direct limits are generalized coproducts and inverse limits are generalized products. So, what is the intuition for inverse limits? Roughly one thinks about inverse limits as “zooming in”, in a sense, whereas direct limits are “blowing up”. Inverse limits can be analogized to generalized intersections, the same way direct limits can be analogized to generalized unions.

*Inverse Systems*

We begin with the notion of an inverse system which can be thought as the “dual notion” of an inverse system. Namely, suppose we have some preordered set , then an ordered pair (where is a set of left -modules and the are -maps) (note the reversal of the roles of as in the case of a directed system) is called an *inverse system over *(or just an inverse system when the context is clear) if and for all in .

Examples of inverse systems of modules abound as much (if not, in some sense, more often) than directed systems. Consider for example the following:

Let be given the order and , then an inverse system over this set is a diagram of modules of the form

Let be some fixed module and let be a set of submodules. Define a preorder on by if and only if . We then define maps to be the inclusions . This clearly defines an inverse system

Consider any set of modules and define the trivial direct system (as defined before), this is trivially an inverse system over .

Let be a left -module and be a left ideal in . Define for each the module to be equal to (where this notion has been defined before). We define mappings for each by the rule . Note that this actually makes sense since (since ). It’s trivially then that these maps work nicely with composition, i.e. that for . Thus, this defines an inverse system as desired.

*Inverse Limits*

Now that we have defined the notion of inverse systems, we may continue to define the dual notion to direct limits–inverse limits. Namely, let be a preordered set and be an inverse system over . An *inverse limit *over this inverse system is an ordered pair where is some fixed module and is a set, called an *(inverse) cone*, of homomorphisms with whenever and universal with respect to this property. In other words, given any set of maps such that there exists a unique with .

As to be expected at this point, inverse limits are unique up to isomorphism:

**Theorem: ***Let be a preordered set and and two direct limits of some inverse system , then .*

**Proof: **By virtue of the existence of maps and which satisfy the universal properties we get maps and such that and . From this we deduce that and . But, since and are also maps satisfying this, we may conclude by uniqueness that and respectively. Thus, is an isomorphism and the conclusion follows.

*Examples of Inverse Limits*

We now look at what the inverse limits are of the inverse systems we previously defined.

For our first example consider the module . It’s easy to see that with the natural projections , and the map is an inverse limit for this inverse system. This is called a *pull-back* of the diagram associated to the inverse system. These, and their direct limit analogues, will have their own post soon enough–they are of the utmost importance.

Suppose now that we have an inverse system of submodules as we had in our second example. We claim that is an inverse limit of this system with the usual inclusions . Indeed, it’s clear that since this merely says that including and then including is the same thing as including , which is trivially true. Thus, to show that really is an inverse limit we need to show that given any left -module and any set such that then there exists a unique such that . But, it’s clear that this is true. Namely, we can define by taking where is any index such that .

What we claim is that when we put the trivial preorder on a collection of modules, the product of the modules is an inverse limit. But, this is clear enough since, by definition, we have the projection maps and since the only maps to check compatibility against are we obviously have that the ‘s are compatible. The universality then follows by the universality of the product of a set of modules.

This last example we leave empty, for we shall discuss a result when we discuss the inverse limit of rings that is better.

**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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