Polynomial identities and noncommutative versal torsors

Abstract

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