Tag - Brauer groups

Susanne Pumpluen: A way to generalize classical results from central simple algebras to the nonassociative setting

11th December, 2023

Recently, the theory of semiassociative algebras and their Brauer monoid was introduced by Blachar, Haile, Matri, Rein, and Vishne as a canonical generalization of the theory of associative central simple algebras and their Brauer group: together with the tensor product semiassociative algebras over a field form a monoid that contains the classical Brauer group as its unique maximal subgroup. We present classes of semiassociative algebras that are canonical generalizations of classes of certain central simple algebras and explore their behavior in the Brauer monoid. Time permitting, we also discuss some - hopefully interesting - particularities of this newly defined Brauer monoid.

Daniel Huybrechts: Brauer groups and twisted sheaves on K3 surfaces

11th September, 2023

This will be a gentle introduction into Brauer groups and twisted sheaves. The emphasis will be on geometric aspects and eventually on moduli spaces of twisted sheaves on K3 surfaces. We will study the different ways to think about Brauer groups as groups of Azumaya algebras, Brauer-Severi varieties, twisted sheaves, 𝔾m-gerbes... How to translate from one to the other, how to define Chern classes, how to split Brauer classes, etc.

Andrei Rapinchu: On the notion of genus for division algebras and algebraic groups

2nd November, 2017

Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) of D as the set of classes [D'] in the Brauer group Br(K) where D' is a central division K-algebra of degree n having the same isomorphism classes of maximal subfields as D. I will review the results on gen(D) obtained in the last several years, in particular the finiteness theorem for gen(D) when K is finitely generated of characteristic not dividing n. I will then discuss how the notion of genus can be extended to arbitrary absolutely almost simple algebraic K-groups using maximal K-tori in place of maximal subfields, and report on some recent progress in this direction. (Joint work with V. Chernousov and I. Rapinchuk)

Laurent Fargues: From local class field theory to the curve and vice versa

29th October, 2015

I will speak about results contained in my article "G-torseurs en théorie de Hodge p-adique" linked to local class field theory. I will in particular explain the computation of the Brauer group of the curve and why its fundamental class is the one from local class field theory.