On Extensions of Rational Modules

TL;DR

This paper characterizes when categories of rational modules over certain algebraic structures are closed under extensions, linking topological and homological conditions, and explores implications in coalgebra theory.

Contribution

It provides a complete characterization of extension closure for rational modules, connecting topological and homological aspects, and generalizes previous partial results in coalgebra theory.

Findings

Characterization of extension closure in rational modules

Connections established with coreflexive coalgebras

Identification of classes where closure holds or fails

Abstract

We investigate when the categories of all rational $A$-modules and of finite dimensional rational modules are closed under extensions inside the category of $C^*$-modules, where $C^*$ is the cofinite topological completion of $A$. We give a complete characterization of these two properties, in terms of a topological and a homological condition. We also give connections to other important notions in coalgebra theory such as coreflexive coalgebras. In particular, we are able to generalize many previously known partial results and answer some questions in this direction, and obtain large classes of coalgebras for which rational modules are closed under extensions as well as various examples where this is not true.