The Brauer group of a cocommutative coalgebra
Let \(R\) be a cocommutative coalgebra. The authors define the notion of an \(R\)-coalgebra \(D\), and an Azumaya \(R\)-coalgebra. To do this, they develop definitions of the cocenter of a coalgebra, the cocommutator coalgebra of a coalgebra map, and of coseparability of a coalgebra. While coseparability had been considered by various authors, the approach here is the first one related to Morita-Takeuchi equivalence. Then an \(R\)- coalgebra \(D\) is called Azumaya if it is \(R\)-cocentral (its cocenter is \(R\)) and \(R\)-coseparable. An equivalent definition involves \(D\) being a quasi-finitely injective cogenerator in an appropriate Morita-Takeuchi context. There is a one-to-one correspondence between subcoalgebras of \(D\) and those of \(R\), and any coalgebra automorphism of \(D\) is inner. The authors construct the Brauer group \(B(R)\) as equivalence classes of Azumaya \(R\)-coalgebras. The equivalence is in terms of quasi-finitely injective cogenerators, but turns out to be a Morita-Takeuchi equivalence relation. If \(R\) is finite-dimensional, then \(B(R)\) is isomorphic to \(B(R^*)\), the usual Brauer group of the dual algebra \(R\). Examples show that this is not always true if \(R\) is infinite-dimensional. However, they show that if \(R\) is coreflexive and irreducible, then \(\text{Br} (R)\) is a subgroup of \(\text{Br} (R_0)\), \(R_0\) the coradical of \(R\).
scientific article; zbMATH DE number 823517