Galois correspondences for Hopf bigalois extensions
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.
scientific article; zbMATH DE number 1211654
scientific article