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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?

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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 $\varphi \in \text{Hom}_R(M, N)$.

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  • $\begingroup$ 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

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