The Generating Condition for Coalgebras

TL;DR

This paper investigates the conditions under which coalgebras generate their comodules, providing counterexamples and conditions for when the generating property holds, thus advancing understanding of coalgebra structure and module theory.

Contribution

It demonstrates that the generating property does not imply self-projectivity in coalgebras and introduces methods to embed any coalgebra into one that generates its comodules.

Findings

Any coalgebra can be embedded into a coalgebra that generates its right comodules.

Counterexamples show the generating property does not imply self-projectivity.

Conditions like being right semiperfect or having finite coradical filtration ensure the generating property.

Abstract

For a ring $R$, the properties of being (left) selfinjective or being cogenerator for the left $R$-modules do not imply one another, and the two combined give rise to the important notion of PF-rings. For a coalgebra $C$, (left) self-projectivity implies that $C$ is generator for right comodules and the coalgebras with this property were called right quasi-co-Frobenius; however, whether the converse implication is true is an open question. We provide an extensive study of this problem. We show that this implication does not hold, by giving a large class of examples of coalgebras having the "generating property". In fact, we show that any coalgebra $C$ can be embedded in a coalgebra $C_\infty$ that generates its right comodules, and if $C$ is local over an algebraically closed field, then $C_\infty$ can be chosen local as well. We also give some general conditions under which the implication "$C$-projective (left) $\Rightarrow C$ generator for right comodules" does work, and such conditions are when $C$ is right semiperfect or when $C$ has finite coradical filtration.