The Brauer group of irreducible coalgebras (Q5936924)

The Brauer group \(\text{Br}(C)\) of a cocommutative coalgebra \(C\) was defined by \textit{B. Torrecillas, F. Van Oystaeyen} and \textit{Y. H. Zhang} [J. Algebra 177, No. 2, 536--568 (1995; Zbl 0837.16037)], where it was shown that \(\text{Br}(C)\) is an Abelian group which is not necessarily torsion. In the same paper it was conjectured that for irreducible \(C\), \(\text{Br}(C)\) is a torsion group. The main aim of the paper under review is to prove this conjecture. In fact the authors show that for cocommutative irreducible \(C\), \(\text{Br}(C)\) can be embedded in the Brauer group of the dual algebra \(C^*\), which is known to be torsion. The method of proof is to study a relation between Morita theory for the algebra \(C^*\) and Morita-Takeuchi theory for the coalgebra \(C\). A localization theoretic argument in terms of the left linear topology of all cofinite closed left ideals is used. The authors also study the connection between \(\text{Br}(C)\) and \(\text{Br}(C_0)\), where \(C_0\) is the coradical of \(C\). Several natural subgroups of the Brauer group of a cocommutative coalgebra are investigated.