Azumaya Objects in Triangulated Bicategories

We introduce the notion of Azumaya object in general homotopy-theoretic settings. We give a self-contained account of Azumaya objects and Brauer groups in bicategorical contexts, generalizing the Brauer group of a commutative ring. We go on to describe triangulated bicategories and prove a characterization theorem for Azumaya objects therein. This theory applies to give a homotopical Brauer group for derived categories of rings and ring spectra. We show that the homotopical Brauer group of an Eilenberg-Mac Lane spectrum is isomorphic to the homotopical Brauer group of its underlying commutative ring. We also discuss tilting theory as an application of invertibility in triangulated bicategories.Comment: 23 pages; final version; to appear in Journal of Homotopy and Related Structure

QUADRATIC SPACES WITH TRIVIAL ARF INVARIANT

93226729

On cubic etale algebras

We describe explicitly a group structure on isomorphism classes of cubic etale algebras with prescribed discriminant. The case of trivial discriminant corresponds to the group of cubic cyclic algebras.6728129

Totally decomposable symplectic and unitary involutions

We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results are new in characteristic 2, otherwise were shown in [3] and [6] respectively.Comment: 11 page