nLab coextension of scalars




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


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

r Rmf(r) Sm r\cdot_R m \coloneqq f( r )\cdot_S m

defines a left action of RR on MM. This construction extends to a functor called restriction of scalars along ff and denoted

f *:SModRMod f^\ast \colon SMod \longrightarrow RMod

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

f *:RModSMod f_* \colon RMod \longrightarrow SMod

Concretely, for any left RR-module MM, f *(M)f_*(M) is the hom set RMod(S,M)RMod(S,M) made into a left SS-module by

(sg)(s)g(ss) (s g)(s') \coloneqq g(s s')

for gRMod(S,M)g \in RMod(S,M) and s,sSs,s' \in S. Here SS is made into a left RR-module by

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


Here is one special case.

For RR a ring, write RRMod for its category of modules. Write Ab = \mathbb{Z}Mod for the category of abelian groups.

Write U:RModAbU\colon R Mod \to Ab for the forgetful functor that forgets the RR-module structure on a module NN and just remembers the underlying abelian group U(N)U(N).


The functor U:RModAbU\colon R Mod \to Ab has a right adjoint

R *:AbRMod R_* : Ab \to R Mod

given by sending an abelian group AA to the abelian group

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

equipped with the RR-module struture by which for rRr \in R an element (U(R)fA)U(R *(A))(U(R) \stackrel{f}{\to} A) \in U(R_*(A)) is sent to the element rfr f given by

rf:rf(rr). r f : r' \mapsto f(r' \cdot r) \,.

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

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

ϵ N:NR *(U(N)) \epsilon_N : N \to R_*(U(N))

is the RR-module homomorphism

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

given on nNn \in N by

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


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 HH is a subgroup of finite index of a group GG, for any field kk there is an inclusion of group algebras

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

and coextension of scalars along ii is naturally isomorphic to extension of scalars along ii.

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.


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

Last revised on November 11, 2023 at 13:55:59. See the history of this page for a list of all contributions to it.