Comparing self-dual corings, Frobenius corings and Ringel self-duality

Abstract

Self-dual corings and self-dual algebra extensions are classified and these (self-)dualities are compared with Ringel (self-)duality, revealing fundamental differences. To each coring $\mathcal{C}$, two algebras are associated, known as the left and the right dual algebra of $\mathcal{C}$. It is shown that these two algebras coincide in a natural way if and only if $\mathcal{C}$ is a Frobenius coring. Various other equivalent characterisations of $\mathcal{C}$ being self-dual are given, in terms of certain algebra extensions being Frobenius extensions and in terms of certain forgetful or restriction functors being Frobenius functors. Ringel self-duality however is shown to be rather different, for which a homological explanation is given.

29 pages