The Dickson Subcategory Splitting Conjecture for Pseudocompact Algebras

Let $A$ be a pseudocompact (or profinite) algebra, so $A=C^{\ast }$ 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.