restriction of scalars




Let f:RSf: R\to S be a homomorphism of algebraic objects such as rings. Let S\cdot_S be an action of SS on an object MM, then by r Rmf(r) Smr\cdot_R m \coloneqq f( r )\cdot_S m is defined an action of RR on MM.

It follows that we have a functor ρ f:SModRMod\rho_f:SMod\to RMod sending S\cdot_S to R\cdot_R which is a forgetful functor.

Adjointly we obtain a functor ϵ f:= RS:RModSMod\epsilon_f:=\otimes_R S:RMod\to SMod called the extension of scalars (see there for more) since for an RR-module MM and the RR-module SS we have that M RSM\otimes_R S is a well defined tensor product of RR modules which becomes an SS module by the operation of SS on itself in the second factor of the tensor. We have an adjunction (e fρ f)(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.