Brauer groups of H-dimodule algebras and truncated power series Hopf algebras.

Abstract

The Brauer group of H-dimodule algebras, BD(R, H), consists of equivalence classes of H-Azumaya algebras; where H is a Hopf algebra over a commutative ring R, and an H-dimodule algebra A is defined to be H-Azumaya if certain maps (analogous to the usual map A (CRTIMES) A (--->) End(A)) are isomorphisms. It is shown that if R is a separably closed field of characteristic p and H is a truncated power series Hopf algebra then a necessary and sufficient condition for an H-Azumaya algebra A to be R-Azumaya (the usual Azumaya R-algebra) is that it be semisimple. An example is given to show that semisimplicity is necessary for this to be true. BD(, 0)(R, H) is the subset of BD(R, H) consisting of only those H-Azumaya algebras that are already R-Azumaya. If each element {A} in BD(, 0)(R, H) has the property that A (TURNEQ) End(V) as an H-module algebra for some finitely generated projective H-module V, then BD(, 0)(R, H) is a subgroup of BD(R, H). For the truncated power series Hopf algebra (alpha)(, p) = k{x}/(x('p)), with x primitive, BD(, 0)(R, (alpha)(, p)) = R* when R is a perfect field of characteristic p and has trivial Brauer group.