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.
Literature:
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