Inverse Limit of Rings (Pt. I)
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.
Inverse System of Rings
Let be a preordered set. Then, an inverse system of rings over is an ordered pair where the are rings and the ring homomorphisms (unital if all the are) satisfying and for any with .
Now, let’s take a look at some examples of inverse systems of rings:
Let and define a preorder on subject to . Then, an inverse system of rings over is a set of rings and a set of maps described in the following diagram:
Let be be a ring (most often a UFD or a PID) we can then define a relation on the set by if and only if . Then, we have canonical maps whenever given by reduction modulo . Said more directly, we map to –this is (not to insult anyone’s intelligence) well-defined for if then so that . Evidently, and for all in . Thus, we see that is an inverse system over
Note that the previous example can easily be generalized. Namely, given any subset we can define the divisibility relation on and create a similar inverse system over . An interesting example is when we take to be a chain. In particular, fix some (where, for the sake of niceness, we assume that is unital, but of course this is unnecessary) and consider we then have natural maps whenever .
In a slightly different (although clearly a more general case of the previous paragraph) direction we consider, given a ring and some ideal or the rings for each . This evidently gives us for each a ring homomorphism given by reduction modulo (recall that ). Clearly one has that and whenenver . Thus, we see that we have constructed an inverse system over the directed set .
Let and define the trivial order on . We then get the trivial inverse system over with just the maps
Inverse Limit of Rings
We now attempt to define the inverse limit of an inverse system of rings. In particular, given a preoredered set and an inverse system over we define an inverse limit of this system to be an ordered pair where is a ring and are ring homomorphisms such that whenever and which satisfy the following universal property: given any ring and any set of ring homomorphisms such that there exists unique ring homomorphism such that .
As always, the inverse limit of an inverse system of rings is unique up to unique isomorphism:
Theorem: Let be a preordered set and an inverse system of rings over . Then, if and are two inverse limits of this inverse system, then .
Proof: By the existence of the maps and we are afforded maps and such that and . From this we may conclude that and –but since and also satisfy this we may conclude by uniqueness that and and so is an isomorphism.
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