nLab
restriction of scalars

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Definition

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

ρ f:SModRMod \rho_f \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

ϵ f() RS:RModSMod \epsilon_f \;\coloneqq\; (-) \otimes_R S \;\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 (ϵ fρ f)(\epsilon_f \dashv \rho_f).

Last revised on July 11, 2021 at 04:59:27. See the history of this page for a list of all contributions to it.