Homomorphisms of topological rings and change-of-scalar functors
Leonid Positselski
Abstract
We consider homomorphisms of complete, separated right or two-sided linear topological rings with countable bases of neighborhoods of zero $\mathfrak f\colon\mathfrak R\to\mathfrak S$. Taut maps of right linear topological rings, strongly right taut maps of two-sided linear topological rings, left proflat continuous ring maps, and topological ring epimorphisms are discussed. For a left proflat topological ring epimorphism $\mathfrak f$, we show that the functor of restriction of scalars on the categories of left contramodules $\mathfrak f_\sharp\colon\mathfrak S{-}\mathsf{Contra}\longrightarrow\mathfrak R{-}\mathsf{Contra}$ is fully faithful. Assuming that the contramodule-to-module forgetful functor $\mathfrak R{-}\mathsf{Contra}\longrightarrow\mathfrak R{-}\mathsf{Mod}$ is fully faithful and the topological ring map $\mathfrak f$ is left proflat, we prove that the commutative square of forgetful functors between the left contramodule and module categories over $\mathfrak S$ and $\mathfrak R$ is a pseudopullback diagram. This provides a description of the essential image of $\mathfrak f_\sharp$ under the conjunction of the respective assumptions. The left adjoint functor to $\mathfrak f_\sharp$ always exists, but is not exact even when $\mathfrak f$ is (pro)flat. A right adjont functor to $\mathfrak f_\sharp$ does not always exist, but for a left proflat map $\mathfrak f$ we construct it explicitly and show that it has good exactness properties. This work is motivated by the theory of contraherent cosheaves of contramodules on formal schemes.