# nLab restriction of scalars

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

Let $f: R\to S$ be a homomorphism of algebraic objects such as rings. Let $\cdot_S$ be an operation of $S$ on an object $M$, then by $r\cdot_R m:=f( r )\cdot_S m$ is defined an action of $R$ on $M$.

It follows that we have a functor $\rho_f:SMod\to RMod$ sending $\cdot_S$ to $\cot_R$ 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 $(\epsilon_f\dashv\rho_f)$.

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