Weakly Azumaya Algebras

Azumaya algebras and derivations

Motives of Azumaya Algebras

We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson-Lichtenbaum conjecture, we apply our results to show the vanishing of SK2(A) for a central simple algebra A of square-free index (prime to the characteristic). This pro...

Deformations of Azumaya Algebras

In this paper we compute the deformation theory of a special class of algebras, namely of Azumaya algebras on a manifold (C or complex analytic). Deformation theory of associative algebras was initiated by Gerstenhaber in [G]. A deformation of an associative algebra A over an Artinian ring a is an a-linear associative algebra structure on A⊗ a such that, for the maximal ideal m of a, A ⊗ m is a...

Azumaya Algebras and Canonical Components

Let M be a compact 3-manifold and Γ = π1(M). The work of Thurston and Culler–Shalen established the SL2(C) character variety X(Γ) as fundamental tool in the study of the geometry and topology of M . This is particularly so in the case when M is the exterior of a hyperbolic knot K in S. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand a...

Evaluating Azumaya algebras on cubic surfaces

Let X be a cubic surface over a local number field k. Given an Azumaya algebra on X, we describe the local evaluation map X(k)→ Q/Z in two cases, showing a sharp dependence on the geometry of the reduction of X. This extends results of Colliot-Thélène, Kanevsky and Sansuc.