Corrigendum to ``A Schneider type theorem for Hopf algebroids''. (Q1018413)

Let \(H\) be a Hopf algebra, and \(A\) a right \(H\)-comodule algebra. \textit{H.-J. Schneider}'s Theorem [Isr. J. Math. 72, No. 1/2, 167-195 (1990; Zbl 0731.16027)] asserts that -- under certain conditions -- surjectivity of the canonical map \(\text{can}\colon A\otimes_{A^{\text{co\,}H}}A\to A\otimes H\) implies its bijectivity. The purpose of this paper is to present a result of this type in the setting of Hopf algebroids. This goes in different steps. First, a generalization of the notion of separable functor, called relative separable functor, is presented. Basic properties of separable functors, such as Maschke's Theorem and Rafael's Theorem are generalized to relative separable functors. Now let \(\mathcal C\) and \(\mathcal D\) be corings over the rings \(A\) and \(L\). If \(\mathcal C\) is a \((\mathcal{C,D})\)-bicomodule, with the left regular \(\mathcal C\)-coaction, then \(\mathcal D\) is called an extension of \(\mathcal C\). Then we can define a functor \(\mathbf R\colon\mathcal M^{\mathcal C}\to{\mathcal M}^{\mathcal D}\), and the authors discuss when the forgetful functor \(\mathbf V^{\mathcal D}\colon\mathcal M^{\mathcal C}\to{\mathcal M}_L\) is \((\mathcal M^{\mathcal D},\mathbf R)\)-separable. Coring extensions can be constructed from entwining structures of rings and corings, and the authors apply their results to this case. Then they turn to Hopf algebroids. Over a Hopf algebroid \(H\), we can define right comodule algebras. To such a right \(H\)-comodule algebra, we can associate an entwining structure, and the above results can be applied. This leads to the main Theorem 5.7. In the particular situation where the Hopf algebroid comes from a Hopf algebra over a field, Schneider's Theorem appears as a special case. In an appendix, the authors collect the results about corings, Hopf algebroids and entwining structures that are needed throughout the paper.