nLab restriction of scalars

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

$$f^\ast \colon SMod \longrightarrow RMod$$

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

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

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 $f_! \dashv f^*)$.

Not only is restriction of scalars a right adjoint, it is also a monadic functor. This can be shown using the monadicity theorem or by direct computation.

Furthermore, not only is restriction of scalars a right adjoint, it is also a left adjoint. That is, it has a right adjoint of its own, called coextension of scalars:

$$f_* \; \coloneqq\; RMod(S, -) \;\colon\; RMod \longrightarrow SMod$$

Related concepts

• extension of scalars$\dashv$ restriction of scalars $\dashv$ coextension of scalars

• restriction (double category theory)