On two finiteness conditions for Hopf algebras with nonzero integral

TL;DR

This paper proves that co-Frobenius Hopf algebras have finite coradical filtration and bounded indecomposable injective comodules, confirming a conjecture and exploring categorical properties of these algebraic structures.

Contribution

It establishes the finiteness of the coradical filtration for co-Frobenius Hopf algebras and extends results to Frobenius tensor categories, also constructing examples that answer previous open questions.

Findings

Coradical filtration of co-Frobenius Hopf algebras is finite

Bounded composition length of indecomposable injective comodules

Existence of co-Frobenius Hopf algebras not of finite type over their Hopf socles

Abstract

A Hopf algebra is co-Frobenius when it has a nonzero integral. It is proved that the composition length of the indecomposable injective comodules over a co-Frobenius Hopf algebra is bounded. As a consequence, the coradical filtration of a co-Frobenius Hopf algebra is finite; this confirms a conjecture by Sorin D\u{a}sc\u{a}lescu and the first author. The proof is of categorical nature and the same result is obtained for Frobenius tensor categories of subexponential growth. A family of co-Frobenius Hopf algebras that are not of finite type over their Hopf socles is constructed, answering so in the negative another question by the same authors.