Point of Post: This is a continuation of this post.
Point of Post: In this post we define the category of directed/inverse system of modules over a given ring and then discuss how the construction of the direct/inverse limit is a functor from this category to the category of -modules. We will then show how this functor is “exact” in the sense that it preserves exact sequences.
We’d now like to take a more theoretical approach to looking at direct/inverse systems, and their limits. Roughly what we we shall see is that direct systems over a fixed preordered set form a nice little category and limits shall be functors on this category. Moreover, what we shall see is that there is a natural notion of exactness for chains of direct/inverse systems, and we shall see that the limit functor preserves exactness when passed into the target category (which will be just ). The reason for this abstraction is nothing more than a desire to put phrase a common construction in the convenient language of category theory. In fact, not only will we see that this phrasing will be useful in and of itself, but it shall serve as a prime example of a more general categorical construct in the future, and this functor shall even occupy much of our time once we start talking about homological algebra (or the left derivation of this functor).
Point of Post: In this post we discuss the notion of the inverse limit of rings, defining inverse systems, defining inverse limits in terms of universal characterizations, and then showing that inverse limits always exist.
As we have already seen it’s easy to go from constructions involving limits of modules to constructions involving limits of rings by just changing “-map” to “ring homomorphism” and “submodule” to “ideal”. This continues in this post where we discuss the notion of inverse limits of inverse systems of rings. The motivation, and the key ideas are the same as they were when we discussed the inverse limits of modules, and so we shall omit motivating remarks and the finer details of proofs since they are almost verbatim, the same.