# nLab restriction of scalars

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

Let $f \colon R \to S$ be a homomorphism of algebraic objects such as rings. Let $\cdot_S$ be an action of $S$ on a module $M$, then

$r\cdot_R m \coloneqq f( r )\cdot_S m$

defines an action of $R$ on $M$. This construction extends to a functor

$\rho_f \colon SMod \longrightarrow RMod$

between categories of modules, sending $\cdot_S$ to $\cdot_R$. This is called restriction of scalars (along $f$).

This functor has a left adjoint functor

$\epsilon_f \;\coloneqq\; (-) \otimes_R S \;\colon\; RMod \longrightarrow SMod$

called extension of scalars, since for an $R$-module $M$ and an $R$-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 $(\epsilon_f \dashv \rho_f)$.

Last revised on July 11, 2021 at 08:59:27. See the history of this page for a list of all contributions to it.