# Definition of pushforward and pullback for modules

Let $R$ be a commutative ring with unit and $\sigma:R\longrightarrow R$ an endomorphism of the ring $R$. Let $M$ be an $R$-module.

1. What are the definitions of $\sigma^*M$ and $\sigma_* M$?
2. How do they behave under tensor product?

Many thanks!

## 1 Answer

More generally, if $\sigma:R\to S$ is a homomorphism, and $N$ is an $S$-module, then the pullback $\sigma^*N$ is an $R$-module where the multiplication is given by $r\cdot_Rn = \sigma(r)\cdot_S n$.

I don't know that there is a nice push-forward, even for an endomorphism $R\to R$. Say $R = \Bbb Z^2$ and $\sigma(a, b) = (a, 0)$. What would be a natural way to define product by $(0,1)\in R$ on $\sigma_*M$?

• Oh right, maybe should I assume $\sigma$ bijective in the pushforward case? and morphism of unital ring. – Stabilo Sep 20 '18 at 9:07
• @Stabilo In which case it will still be the pullback of the inverse morphism. So pullbacks are nice, pushforwards not so much. – Arthur Sep 20 '18 at 9:40
• Thank you very much! – Stabilo Sep 20 '18 at 10:07