Equivariant maps for modules over different rings

Question

This is probably well known, but I'm trying to understand "equivariant maps" over different rings. Namely, let $R,S$ be rings and let $\phi:R\to S$ be a ring homomorphism. Let $M$ be an $R$-module with action $\rho:R\otimes M\to M$, and $M'$ an $S$-module with $\rho':S\otimes M'\to M'$. Then consider maps $f:M\to M'$ such that the following diagram commutes: $$ \require{AMScd} \begin{CD} R\otimes M @> \rho > > M\\ @V \phi\otimes f V V @V V f V\\ S\otimes M' @> > \rho' > M' \end{CD} $$

For $R=S$, $\phi=\text{id}_R$, we get regular intertwiners for modules over $R$; but one could as well consider some other non-trivial isomorphism of $R$. For $M=M'$ we could think about base change from $S$-mod to $R$-mod, but those don't seem to have anything to do with such an equivariance.

Question:

Are these non-trivial and/or interesting objects? How do you call them, and where can I know more about them?

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Just came across slight variations of this question:

• Defining maps from modules over different rings

• A twisted version of a module homomorphism

Answer 1

These maps are called $R$-module homomorphisms, where $M'$ is regarded as an $R$-module via restriction along the homomorphism $\phi$. They are of course non-trivial and very interesting objects, in general.

For instance, Frobenius reciprocity is the statement that the obvious map $$\mathrm{Hom}_R(M, \mathrm{Res}(M')) \to \mathrm{Hom}_S(S \otimes_R M, M')$$ is an isomorphism. The domain of this map is the set of maps you are looking at.