01165nas a2200109   4500008004100000245004700041210004700088260001200135520085700147100001401004856003701018       2020                        eng d00aTwisted Ehresmann Schauenburg bialgebroids0 aTwisted Ehresmann Schauenburg bialgebroids  c09/20203 aWe construct an invertible normalised 2 cocycle on the Ehresmann Schauenburg bialgebroid of a cleft Hopf Galois extension under the condition that the corresponding Hopf algebra is cocommutative and the image of the unital cocycle corresponding to this cleft Hopf Galois extension belongs to the centre of the coinvariant subalgebra. Moreover, we show that any Ehresmann Schauenburg bialgebroid of this kind is isomorphic to a 2-cocycle twist of the Ehresmann Schauenburg bialgebroid corresponding to a Hopf Galois extension without cocycle, where comodule algebra is an ordinary smash product of the coinvariant subalgebra and the Hopf algebra (i.e. $\C(B/#_{\sigma}H, H)\simeq \C(B\#H, H)^{\tilde{\sigma}}$). We also study the theory in the case of a Galois object where the base is trivial but without requiring the Hopf algebra to be cocommutative.1 aHan, Xiao  uhttps://arxiv.org/abs/2009.02764