symmetric monoidal (∞,1)-category of spectra
Let $f \colon R \to S$ be a homomorphism of algebraic objects such as rings. Let $\cdot_S$ be an action of $S$ on a module $M$, then
defines an action of $R$ on $M$. This construction extends to a functor
between categories of modules, sending $\cdot_S$ to $\cdot_R$. This is called restriction of scalars (along $f$).
This functor has a left adjoint functor
called extension of scalars, since for an $R$-module $M$ and an $R$-module $S$ we have that $M\otimes_R S$ is a well defined tensor product of $R$ modules which becomes an $S$ module by the operation of $S$ on itself in the second factor of the tensor. We have an adjunction $f_! \dashv f^*)$.
Not only is restriction of scalars a right adjoint, it is also a monadic functor. This can be shown using the monadicity theorem or by direct computation.
Furthermore, not only is restriction scalars a right adjoint, it is also a left adjoint. That is, it has a right adjoint of its own, called coextension of scalars:
Last revised on May 12, 2023 at 15:59:36. See the history of this page for a list of all contributions to it.