Homomorphisms of topological rings and change-of-scalar functors

Abstract

We consider homomorphisms of complete, separated right or two-sided linear topological rings with countable bases of neighborhoods of zero $\mathfrak{f}:\mathfrak{R}\to \mathfrak{S}$. 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 ${\mathfrak{f}}_{♯}:\mathfrak{S}-\mathsf{Contra}⟶\mathfrak{R}-\mathsf{Contra}$ is fully faithful. Assuming that the contramodule-to-module forgetful functor $\mathfrak{R}-\mathsf{Contra}⟶\mathfrak{R}-\mathsf{Mod}$ 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 ${\mathfrak{f}}_{♯}$ under the conjunction of the respective assumptions. The left adjoint functor to ${\mathfrak{f}}_{♯}$ always exists, but is not exact even when $\mathfrak{f}$ is (pro)flat. A right adjont functor to ${\mathfrak{f}}_{♯}$ 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.

LaTeX 2e with mathrsfs and xy-pic; 55 pages, 15 commutative diagrams; v.2: Examples 9.8 added, several paragraphs inserted here and there, many misprints corrected, references added