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 = $\mathbb{Z}$Mod for the category of abelian groups.

Write $U\colon R Mod \to Ab$ 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 $U\colon R Mod \to Ab$ has a right adjoint

$R_* : Ab \to R Mod$

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

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

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 $r f$ given by

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

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

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

$\epsilon_N : N \to R_*(U(N))$

is the $R$-module homomorphism

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

given on $n \in N$ by

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

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)