Galois correspondences for Hopf bigalois extensions
type: Item
instance of: scholarly article
title: Galois correspondences for Hopf bigalois extensions
zbMATH Open document ID: 0912.16019
author: Peter Schauenburg
published in: Journal of Algebra
publication date: 1999-05-25T00:00:00Z
review text: The fundamental theorem of Hopf Galois theory concerns a Hopf algebra \(H\) and a faithfully flat right \(H\)-Galois extension \(A\) of \(k\) (a commutative ring). Then there exists a unique Hopf algebra \(L=L(H,A)\) and left \(L\)-comodule algebra structure on \(A\) such that \(A\) is a left \(L\)-Galois extension of \(k\) and an \(L\)-\(H\) bicomodule (i.e., \(A\) is an \(L\)-\(H\) bigalois extension). The fundamental theorem gives correspondences between the coideal left ideals in \(L\) and the \(H\)-subcomodule algebras of \(A\) [see \textit{F. Van Oystaeyen} and \textit{Y. Zhang}, \(K\)-Theory 8, No. 3, 257-269 (1994; Zbl 0814.16034)] which form a Galois connection. In the paper under review, the author proves a version of this correspondence theorem when \(H\) has a bijective antipode. The Galois connection between quotient coalgebras and left modules of \(L\) and \(H\)-subcomodule algebras of \(A\) induces a bijection between those quotients over which \(L\) is faithfully coflat, and those subalgebras of \(A\) over which \(A\) is faithfully flat. The lattices of Hopf subalgebras of \(L\) and \(H\) over which these are faithfully flat are isomorphic. An example shows that this last result is not true for the lattices of those of their \(k\)-flat quotient Hopf algebras over which they are faithfully coflat.
reviewed by: Earl J. Taft
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 06A15
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 1211654
zbMATH Keywords: Hopf Galois theory
zbMATH Keywords: faithfully flat right Galois extensions
zbMATH Keywords: left comodule algebras
zbMATH Keywords: left Galois extensions
zbMATH Keywords: bicomodules
zbMATH Keywords: bigalois extensions
zbMATH Keywords: Galois connections
zbMATH Keywords: bijective antipodes
zbMATH Keywords: quotient coalgebras
zbMATH Keywords: lattices of Hopf subalgebras
MaRDI profile type: Publication
full work available at URL: jabr.1997.7237
OpenAlex ID: W1965826290
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DOI: 10.1006/JABR.1997.7237
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description: scientific article; zbMATH DE number 1211654