Polynomial Identities and Noncommutative Versal Torsors

To any cleft Hopf Galois object, i.e., any algebra H obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras” AH and U α H . The algebra A α H 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, AH is a cleft H-Galois extension of a “big” commutative algebra B H . Any “form” of H can be obtained from AH by a specialization of B H and vice versa. If the algebra H is simple, then AH is an Azumaya algebra with center B H . The algebra U α H 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 H are satisfied. We construct an embedding of U H into A α H ; this embedding maps the center Z H of U α H into B α H when the algebra H is simple. In this case, under an additional assumption, AH ∼= B H ⊗Zα H U α H , thus turning AH into a central localization of U α H . We completely work out these constructions in the case of the four-dimensional Sweedler algebra.