nLab
coextension of scalars

Contents

Idea

A right adjoint to restriction of scalars. The dual notion to extension of scalars .

Properties

Here is one special case.

For RR a ring, write RRMod for its category of modules. Write Ab = \mathbb{Z}Mod for the category of abelian groups.

Write U:RModAbU\colon R Mod \to Ab for the forgetful functor that forgets the RR-module structure on a module NN and just remembers the underlying abelian group U(N)U(N).

Lemma

The functor U:RModAbU\colon R Mod \to Ab has a right adjoint

R *:AbRMod R_* : Ab \to R Mod

given by sending an abelian group AA to the abelian group

U(R *(A))Ab(U(R),A) U(R_*(A)) \coloneqq Ab(U(R),A)

equipped with the RR-module struture by which for rRr \in R an element (U(R)fA)U(R *(A))(U(R) \stackrel{f}{\to} A) \in U(R_*(A)) is sent to the element rfr f given by

rf:rf(rr). r f : r' \mapsto f(r' \cdot r) \,.

This is called the coextension of scalars along the ring homomorphism R\mathbb{Z} \to R.

The unit of the (UR *)(U \dashv R_*) adjunction

ϵ N:NR *(U(N)) \epsilon_N : N \to R_*(U(N))

is the RR-module homomorphism

ϵ N:NHom Ab(U(R),U(N)) \epsilon_N : N \to Hom_{Ab}(U(R), U(N))

given on nNn \in N by

j(n):rrn. j(n) : r \mapsto r n \,.

References

  • H. Fausk, P. Hu, Peter May, Isomorphisms between left and right adjoints (pdf)

Revised on September 11, 2013 20:29:15 by Urs Schreiber (145.116.131.249)