# nLab coextension of scalars

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 = $ℤ$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 $U\left(N\right)$.

###### 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

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

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

$rf:r\prime ↦f\left(r\prime \cdot r\right)\phantom{\rule{thinmathspace}{0ex}}.$r f : r' \mapsto f(r' \cdot r) \,.

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

The unit of the $\left(U⊣{R}_{*}\right)$ adjunction

${ϵ}_{N}:N\to {R}_{*}\left(U\left(N\right)\right)$\epsilon_N : N \to R_*(U(N))

is the $R$-module homomorphism

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

given on $n\in N$ by

$j\left(n\right):r↦rn\phantom{\rule{thinmathspace}{0ex}}.$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)