Quiver concomitants are often reflexive Azumaya

There Are Enough Azumaya Algebras on Surfaces

Using Maruyama’s theory of elementary transformations, I show that the Brauer group surjects onto the cohomological Brauer group for separated geometrically normal algebraic surfaces. As an application, I infer the existence of nonfree vector bundles on proper normal algebraic surfaces.

Azumaya Categories

We define the notions of Azumaya category and Brauer group in category theory enriched over some very general base category V. We prove the equivalence of various definitions, in particular in terms of separable categories or progenerating bimodules. When V is the category of modules over a commutative ring R with unit, we recapture the classical notions of Azumaya algebra and Brauer group and ...

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Azumaya Monads and Comonads

The definition of Azumaya algebras over commutative rings R requires the tensor product of modules over R and the twist map for the tensor product of any two R-modules. Similar constructions are available in braided monoidal categories, and Azumaya algebras were defined in these settings. Here, we introduce Azumaya monads on any category A by considering a monad (F,m, e) on A endowed with a dis...

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...