nLab
coextension of scalars

Idea

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

Properties

Here is one special case.

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

Write $U:R\mathrm{Mod}\to \mathrm{Ab}$ for the forgetful functor that forgets the $R$-module structure on a module $N$ and just remembers the underlying abelian group $U(N)$.

Lemma

The functor $U:R\mathrm{Mod}\to \mathrm{Ab}$ has a right adjoint

$$R_{*}:\mathrm{Ab}\to R\mathrm{Mod}$$R_* : Ab \to R Mod

given by sending an abelian group $A$ to the abelian group

$$U(R_{*}(A))≔\mathrm{Ab}(U(R),A)$$U(R_*(A)) \coloneqq Ab(U(R),A)

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

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

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

The unit of the $(Udash{R}_{*})$ adjunction

$${ϵ}_N:N\to R_{*}(U(N))$$\epsilon_N : N \to R_*(U(N))

is the $R$-module homomorphism

$${ϵ}_N:N\to {\mathrm{Hom}}_{\mathrm{Ab}}(U(R),U(N))$$\epsilon_N : N \to Hom_{Ab}(U(R), U(N))

given on $n\in N$ by

$$j(n):r\mapsto rn.$$j(n) : r \mapsto r n \,.

References

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