The splitting problem for coalgebras: a direct approach

TL;DR

This paper provides a direct proof that if the rational part of any right module over a coalgebra splits as a direct summand, then the coalgebra must be finite dimensional, addressing the splitting problem for coalgebras.

Contribution

It offers a new, more straightforward proof of a known result regarding the structure of coalgebras and their modules.

Findings

The rational part of any right module splits as a direct summand only if the coalgebra is finite dimensional.

The proof simplifies previous approaches to the splitting problem.

Supports the characterization of finite dimensional coalgebras through module decomposition.

Abstract

In this note we give a different and direct short proof to a previous result of Nastasescu and Torrecillas in \cite{NT} stating that if the rational part of any right $C^*$ module $M$ is a direct sumand in $M$ then $C$ must be finite dimensional (the splitting problem for coalgebras).