Hopf Galois extensions of Hopf algebroids

We study Hopf Galois extensions of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we introduce (skew-) regular comodules and generalize the structure theorem for relative Hopf modules. Also, we show that if N P is a left L-Galois extension and is a 2-cocycle of L, then for the twisted comodule algebra _P, N_P is a left Hopf Galois extension of the twisted Hopf algebroid L^. We study twisted Drinfeld doubles of Hopf algebroids as examples for the Drinfeld twist theory. Finally, we introduce cleft extension and -twisted crossed products of Hopf algebroids. Moreover, we show the equivalence of cleft extensions, -twisted crossed products, and Hopf Galois extensions with normal basis properties, which generalize the theory of cleft extensions of Hopf algebras.