nLab coextension of scalars

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.

Related concepts

• extension of scalars$\dashv$ restriction of scalars $\dashv$ coextension of scalars

References

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