The Brauer Group of Irreducible Coalgebras

Abstract In this paper we solve a previously formulated conjecture (B. Torrecillas, F. Van Oystaeyen, and Y. H. Zhang, J. Algebra 177 (1995), 568). That is, for a cocommutative irreducible coalgebra C , the homomorphism (−)*:  Br ( C ) →  Br ( C *) is injective. The proof uses Morita–Takeuchi theory and the linear topology of all closed cofinite left ideals in C *. As an inmediate consequence, Br ( C ) is a torsion group. Some cases where the map (−)* is an isomorphism are studied. It is also deduded from the main result that the inclusion of the coradical C 0 into C induces a monomorphism i ∗:  Br ( C ) →  Br ( C 0 ). New examples of Brauer groups of cocommutative coalgebras may be given using this fact.

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