Suppose we have a system of ring homomorphisms $\cdots \to R_{i+1} \stackrel{f_i}{\to} R_i \to \cdots \stackrel{f_0}{\to} R_0$ (we may assume all the maps are injective, but that's not necessary for my needs), and $M,N$ are $R_0$-modules. Via the $f_i$, they are also $R_i$-modules for any $i$, and any $R_i$-linear map between them is also $R_{i+1}$-linear (because $R_{i+1}$ acts via $f_i$ and the action of $R_i$), so we have a system $\operatorname{Hom}_{R_0}(M,N)\to \operatorname{Hom}_{R_1}(M,N) \to \cdots \to \operatorname{Hom}_{R_i}(M,N)\to \cdots$.

The question is then:

Is $\operatorname{colim}_i \operatorname{Hom}_{R_i}(M,N) \simeq \operatorname{Hom}_{\lim_i R_i}(M,N)$?

Naively, the left hand side is "anything that's $R_i$-linear for all $i$ big enough", and the right hand side is, well, the same. It seems a little too nice to be true though, but I'm not sure where (if anywhere) it fails. What subtlety am I missing?


Unfortunately, it's not true. The problem lies in your interpretation of the left hand side; it should rather be "everything that's $R_i$-linear for some $i$".

For a counter example, let $R_i=\mathbf Z[x_i, x_{i+1}, \dots]$ be the polynomial ring in countably many variables labeled by the integers starting with $i$. Let $R_{i+1} \to R_i$ be the inclusion. Then $R:= \varprojlim R_i = \mathbf Z$. An $R_0$-module is an abelian group $M$ with a distinguished collection of endomorphisms $f_0, f_1, \dots$. An element of

$$\varinjlim \text{Hom}_{R_i}(M, N)$$

is a morphism of abelian groups which commutes with the distinguished endomorphisms of sufficiently high degree. On the other hand, an element of $\text{Hom}_R(M, N)$ is just a morphism of abelian groups, without any conditions. In general, there will be many more of these.

However, you are right to say that there is a map $\varinjlim \text{Hom}_{R_i}(M, N) \to \text{Hom}_R(M, N)$. Namely, given a morphism $\varphi \in \text{Hom}_{R_i}(M, N)$, we can take the "restriction of scalars" along the morphism $R\to R_i$ to view it as a morphism $\varphi \in \text{Hom}_R(M, N)$.

  • $\begingroup$ Right, I see where I got confused. Another way to say it would be that $lim R_i$-linear maps don't have to be $R_i$-linear for any $i$, but anything in $colim Hom_{R_i}(-)$ of course is. $\endgroup$ – Ketil Tveiten Jan 30 '14 at 14:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.