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So I have encountered the following task: Let $\phi:R\rightarrow S$ be a ringhomomorphism of commutative rings. Decide whether the functor $$\phi^*: Mod_R\rightarrow Mod_S,M\mapsto M\otimes_R S $$ admits a left/right adjoint.

I know that for a given R-Module N, the functor $$\Phi: Mod_R\rightarrow Mod_R,M\mapsto M\otimes_R N $$ is leftadjoint to the functor $Hom_R(N,-)$. I can further explain that if needed.

Firstly, I dont really get how we can consider $M\otimes_R S$ as an S-Module. I do understand how we act on S (just by multiplication), but I do not understand how one can act on M. It somehow has to be induced by the ringmap i think. Thanks for any help!!

Adronic
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It seems your question is why $M \otimes_R S$ can be given the structure of an $S$-module. But you have already indicated the $S$-scalar multiplication: For $s,t \in S$ and $m \in M$ we define $t(m \otimes s) = m \otimes (ts)$ and extend linearly. This multiplication indeed makes $M \otimes_R S$ into an $S$-module.

If you're still confused, then it might help to think about explicit examples. For instance, how do we naturally make $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z} \cong \mathbb{Q}$ into a $\mathbb{Q}$-module?

Qi Zhu
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  • oh yeah true i thought i have to find a way to act on M but thats just a mistake. My actual question was about the left/right-adjoint though:) – Adronic Mar 31 '22 at 17:50