Polynomial identities and noncommutative versal torsors

AbstractTo any cleft Hopf Galois object, i.e., any algebra Hα obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras”AHα and UHα. The algebra AHα is obtained by twisting the multiplication of H with the most general two-cocycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construction, AHα is a cleft H-Galois extension of a “big” commutative algebra BHα. Any “form” of Hα can be obtained from AHα by a specialization of BHα and vice versa. If the algebra Hα is simple, then AHα is an Azumaya algebra with center BHα. The algebra UHα is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of Hα are satisfied. We construct an embedding of UHα into AHα; this embedding maps the center ZHα of UHα into BHα when the algebra Hα is simple. In this case, under an additional assumption, AHα≅BHα⊗ZHαUHα, thus turning AHα into a central localization of UHα. We completely work out these constructions in the case of the four-dimensional Sweedler algebra