Semiartinian profinite algebras have nilpotent Jacobson radical

TL;DR

This paper proves that semiartinian profinite algebras have a nilpotent Jacobson radical, establishing a link between T-nilpotence and nilpotence, and answering open questions in the field.

Contribution

It introduces a method to analyze the coradical filtration of coalgebras and demonstrates that semiartinian profinite algebras possess a nilpotent Jacobson radical, resolving previous open questions.

Findings

Semiartinian profinite algebras have nilpotent Jacobson radical.

T-nilpotence implies nilpotence for the Jacobson radical in semilocal profinite algebras.

Provides a new approach to study the finiteness of the coradical filtration.

Abstract

We give a method to study the finiteness of the coradical filtration of a coalgebra; as a consequence, we show that a left semiartinian profinite algebra has nilpotent Jacobson radical and is right semiartinian too. Equivalently, we show that a for a semilocal profinite algebra, T-nilpotence implies nilpotence for the Jacobson radical. This answers two open questions from \cite{INT}.