symmetric monoidal (∞,1)-category of spectra
Coextension of scalars is the right adjoint to restriction of scalars. It is the dual notion to extension of scalars.
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
defines a left action of $R$ on $M$. This construction extends to a functor called restriction of scalars along $f$ and denoted
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
Concretely, for any left $R$-module $M$, $f_*(M)$ is the hom set $RMod(S,M)$ made into a left $S$-module by
for $g \in RMod(S,M)$ and $s,s' \in S$. Here $S$ is made into a left $R$-module by
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 $U(N)$.
The functor $U\colon R Mod \to Ab$ has a right adjoint
given by sending an abelian group $A$ to the abelian group
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
This is called the coextension of scalars along the ring homomorphism $\mathbb{Z} \to R$.
The unit of the $(U \dashv R_*)$ adjunction
is the $R$-module homomorphism
given on $n \in N$ by
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
and coextension of scalars along $i$ is naturally isomorphic to extension of scalars along $i$.
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.
Last revised on June 18, 2024 at 19:56:54. See the history of this page for a list of all contributions to it.