On Extensions of Rational Modules

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 $C^{\ast }$-modules, where $C^{\ast }$ 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.

21p

References in corpus (4)

• The Generating Condition for Coalgebras
• Abstract integrals in algebra: coalgebras, Hopf algebras and compact groups
• Chain coalgebras and distributivity
• Localization in coalgebras. Applications to finiteness conditions