Galois extensions for co-Frobenius Hopf algebras (Q1375979)

Let \(H\) be a co-Frobenius Hopf algebra over a field \(k\), \(A\) an \(H\)-comodule algebra, \(A^{\text{co }H}\) the ring of coinvariants. The authors give a list of conditions, each of which is equivalent for \(A/A^{\text{co }H}\) to be an \(H\)-Galois extension. This is done by linking the smash product \(A\#H^{*\text{rat}}\) to \(A^{\text{co }H}\) by a Morita context (\(H^{*\text{rat}}\) is the unique maximal rational submodule of \(H^*\), and \(H\) is co-Frobenius if and only if \(H^{*\text{rat}}\neq 0\)). This paper generalizes and parallels previous results which start with \(H\) being finite-dimensional, in particular [\textit{M. Cohen}, \textit{D. Fischman} and \textit{S. Montgomery}, J. Algebra 133, No. 2, 351-372 (1990; Zbl 0706.16023); \textit{M. Cohen} and \textit{D. Fischman}, ibid. 149, No. 2, 419-437 (1992; Zbl 0788.16029); \textit{Y.-H. Zhang}, Commun. Algebra 20, No. 7, 1907-1915 (1992; Zbl 0771.16013); \textit{H. Chen} and \textit{C. Cai}, ibid. 22, No. 1, 253-267 (1994; Zbl 0844.16020); \textit{Y. Doi}, Lect. Notes Pure Appl. Math. 158, 39-53 (1994; Zbl 0831.16023)].