10 research outputs found
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
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