Homomorphisms of topological rings and change-of-scalar functors

TL;DR

This paper studies homomorphisms of topological rings and their impact on categories of contramodules, establishing conditions for functor properties and adjoint existence relevant to formal schemes.

Contribution

It characterizes when restriction of scalars functors are fully faithful and constructs explicit adjoints under proflatness assumptions.

Findings

Restriction of scalars functor is fully faithful for left proflat epimorphisms.

Under certain conditions, a pseudopullback diagram of forgetful functors is established.

Explicit construction of a right adjoint functor with good exactness properties for left proflat maps.

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.