Q:

Universal Coefficient Theorem - what kind of morphisms?

Let $G$ be an $R$-module, where $R$ is a P.I.D., and let $X$ be a topological space. We have the exact sequence
$$0 \rightarrow H_n(X) \otimes G \rightarrow H_n(X; G) \rightarrow \text{Tor}(H_{n-1}(X), G) \rightarrow 0.$$
The objects in this sequence are $R$-modules.
Question: Are the homomorphisms in this sequence $R$-module homomorphisms or just $\mathbb{Z}$-module homomorphisms?
Remark: Hatcher, Spanier, Google and the other references that I looked up do not explicitly say this, and it's not clear to me.

A:

Yes, the morphisms are $R$-module morphisms. This is very easy to see using some abstract nonsense. Everything here is functorial in $G$, i.e. if $f:G\to G'$ is a $\mathbb{Z}$-module morphism, the induced diagram
$$\begin{array}{c}
H_\ast(X)\otimes G & \rightarrow & H_\ast(X;G) & \rightarrow & Tor(H_{\ast-1}(X),G) \\
   \downarrow      &             &  \downarrow &             &  \downarrow \\
H_\ast(X)\otimes G' & \rightarrow & H_\ast(X;G') & \rightarrow & Tor(H_{\ast-1}(X),G')
\end{array}$$
commutes. Now set $G':=G$ and $f(g):=rg$ for some $r\in R$ and you will get that the maps commute with multiplication by elements of $R$: In other words the maps are $R$-module morphisms.

