The Dickson Subcategory Splitting Conjecture for Pseudocompact Algebras

TL;DR

This paper proves that for pseudocompact algebras, if the semiartinian part of every module splits off, then the algebra is semiartinian, confirming a conjecture of Faith in this context.

Contribution

It establishes a splitting condition that characterizes semiartinian pseudocompact algebras, providing a positive resolution of Faith's conjecture for this class.

Findings

Semiartinian pseudocompact algebras are characterized by splitting of the semiartinian part in modules.

The paper confirms Faith's conjecture for duals of coalgebras.

Splitting of the Dickson subcategory is equivalent to semiartinian property in this setting.

Abstract

Let $A$ be a pseudocompact (or profinite) algebra, so $A=C^*$ where $C$ is a coalgebra. We show that the if the semiartinian part (the "Dickson" part) of every $A$-module $M$ splits off in $M$, then $A$ is semiartinian, also giving a positive answer in the case of algebras arising as dual of coalgebras (pseudocompact algebras), to a well known conjecture of Faith.