Special lecture course of GRK 2240:
Brauer  groups in algebra and geometry

Winter semester 22/23

  • Date:  Thursday, 14:15 - 15:30
  • Place:  Lecturehall 5 in G.10.07 (BUW) or seminar room 25.22-03.73 (HHU)
  • Start:  Thursday, October 27
  • Speaker: Prof. Dr. Stefan Schröer

On Thursdays with full workshop program, the  special lecture course physically takes place in the lecture hall at BUW. On the no-professor Thursdays, we will arrange a video transmission to BUW from the seminar room at HHU. See the schedule


Content:  The Brauer group comprises equivalence classes of so-called Azumaya algebras. It can be formed for fields, rings, or ringed spaces, and gives a  truly foundational invariant having striking applications in various fields of mathematics. To mention   a few: In group theory, they measure the obstruction to pass from a projective representation to a linear representation. In algebraic number theory, they provide an elegant formulation of class field theory. In algebraic geometry, they are directly related to twisted forms of projective n-spaces, and frequently contain obstructions against existence of tautological objects for moduli problems. In complex geometry, they describe the relation between algebraic and transcendental cycles. In arithmetic geometry, they   can be used to explain why certain schemes over numbers fields may or may not contain rational points.

The main reson for this amazing versality is that Br(X) can be expressed in terms of the second cohomology group H^2(X,G_m) formed with respect to the 'étale topology, a truely foundational observation of Grothendieck. The goal of the lecture course is to give an introduction to Brauer groups and their applications. In the second half of the course, we plan to focus on the problem of representing cohomology classes by Azumaya algebras, and applications of Brauer groups to moduli problems and invariant theory.

P. Gille, T. Szamuely: Central simple algebras and Galois cohomology.
J.-L.Colliot-Thélène, A. Skorobogatov: The Brauer-Grothendieck group.
A. Grothendieck: Le groupe de Brauer I-III.
B. Farb, R. Dennis: Noncommutative algebra.

Office hours: 
Prof. Dr. Stefan Schröer: Tuesday from 10:30 - 11:30