Semiartinian Profinite Algebras have Nilpotent Jacobson Radical

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 Iovanov et al. (J Algebra 320(5):2144–2155, 2008).