nLab
restriction of scalars

Contents

Definition

Let f:RSf: R\to S be a homomorphism of algebraic objects such as rings. Let S\cdot_S be an operation of SS on an object MM, then by r Rm:=f(r) Smr\cdot_R m:=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 cot R\cot_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 (ϵ fρ f)(\epsilon_f\dashv\rho_f).

Revised on September 11, 2013 20:29:27 by Urs Schreiber (145.116.131.249)