The Brauer group of a cocommutative coalgebra
type: Item
instance of: scholarly article
title: The Brauer group of a cocommutative coalgebra
zbMATH Open document ID: 0837.16037
author: Yinhuo Zhang
author: Blas Torrecillas
author: Fred Van Oystaeyen
published in: Journal of Algebra
publication date: 1996-04-24T00:00:00Z
review text: 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 \(R\).
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 823517
zbMATH Keywords: cocommutative coalgebras
zbMATH Keywords: Azumaya coalgebras
zbMATH Keywords: cocenter
zbMATH Keywords: cocommutator coalgebras
zbMATH Keywords: coseparability
zbMATH Keywords: quasi-finitely injective cogenerators
zbMATH Keywords: Morita-Takeuchi contexts
zbMATH Keywords: Brauer groups
zbMATH Keywords: Morita-Takeuchi equivalence
reviewed by: Earl J. Taft
MaRDI profile type: Publication
full work available at URL: jabr.1995.1311
OpenAlex ID: W1996249947
DOI: 10.1006/JABR.1995.1311
description: scientific article; zbMATH DE number 823517