symmetric monoidal (∞,1)-category of spectra
Let $f: R\to S$ be a homomorphism of algebraic objects such as rings. Let $\cdot_S$ be an action of $S$ on an object $M$, then by $r\cdot_R m \coloneqq f( r )\cdot_S m$ is defined an action of $R$ on $M$.
It follows that we have a functor $\rho_f:SMod\to RMod$ sending $\cdot_S$ to $\cdot_R$ which is a forgetful functor.
Adjointly we obtain a functor $\epsilon_f:=\otimes_R S:RMod\to SMod$ called the extension of scalars (see there for more) since for an $R$-module $M$ and the $R$-module $S$ we have that $M\otimes_R S$ is a well defined tensor product of $R$ modules which becomes an $S$ module by the operation of $S$ on itself in the second factor of the tensor. We have an adjunction $(e_f\dashv\rho_f)$.
Last revised on April 20, 2020 at 09:24:31. See the history of this page for a list of all contributions to it.