nLab restriction of scalars




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

r Rmf(r) Sm r\cdot_R m \coloneqq f( r )\cdot_S m

defines an action of RR on MM. This construction extends to a functor

f *:SModRMod f^\ast \colon SMod \longrightarrow RMod

between categories of modules, sending S\cdot_S to R\cdot_R. This is called restriction of scalars (along ff).

This functor has a left adjoint functor

f !S R():RModSMod f_! \; \coloneqq\; S \otimes_R (-) \;\colon\; RMod \longrightarrow SMod

called extension of scalars, since for an RR-module MM and an 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 f !f *)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 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 *RMod(S,):RModSMod f_* \; \coloneqq\; RMod(S, -) \;\colon\; RMod \longrightarrow SMod

Last revised on May 12, 2023 at 15:59:36. See the history of this page for a list of all contributions to it.