# nLab coextension of scalars

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

Coextension of scalars is the right adjoint to restriction of scalars. It is the dual notion to extension of scalars.

## Definition

Let $f \colon R \to S$ be a homomorphism of algebraic objects such as rings. If $\cdot_S$ is the left action of $S$ on some module $M$, then

$r\cdot_R m \coloneqq f( r )\cdot_S m$

defines a left action of $R$ on $M$. This construction extends to a functor called restriction of scalars along $f$ and denoted

$f^\ast \colon SMod \longrightarrow RMod$

This functor has both a left adjoint and a right adjoint. The left adjoint $f_!$ is called extension of scalars, which its right adjoint is called coextension of scalars and denoted

$f_* \colon RMod \longrightarrow SMod$

Concretely, for any left $R$-module $M$, $f_*(M)$ is the hom set $RMod(S,M)$ made into a left $S$-module by

$(s g)(s') \coloneqq g(s' s)$

for $g \in RMod(S,M)$ and $s,s' \in S$. Here $S$ is made into a left $R$-module by

$r \cdot s \coloneqq f(r) s$

## Example

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

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

## Properties

### Relation to extension of scalars

In some cases coextension of scalars is naturally isomorphic to extension of scalars, which is the left adjoint to restriction of scalars. For example if $H$ is a subgroup of finite index of a group $G$, for any field $k$ there is an inclusion of group algebras

$i \colon k[H] \to k[G]$

and coextension of scalars along $i$ is naturally isomorphic to extension of scalars along $i$.

### Frobenius extensions

Co-Extension of scalars coincides with extension of scalars (to make an ambidextrous adjunction with restriction of scalars) in the case of Frobenius extensions, see there for more.

## References

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

Last revised on June 18, 2024 at 19:56:54. See the history of this page for a list of all contributions to it.