Brauer group and Severi-Brauer varieties

Seminar at the University of Regensburg in Winter Semester 2022/2023

Wednesday 12:15-13:45 in M101.

The first meeting is on 26.10.

Abstract

The seminar is dedicated to a topic that connects Galois theory, simple (non-commutative) algebras and algebraic geometry.

A central simple algebra over $k$ is a finite-dimensional associative algebra over $k$ with the center equal to $k$ and that has no non-trivial two-sided ideals. Examples of these are Hamilton quaternions over $\mathbb{R}$ and matrix algebras over any base field. One can use tensor multiplication over $k$ to define product of such algebras and form a monoid. After imposing what's known as Morita equivalence one obtains the Brauer group of the base field $k$.

A Severi-Brauer variety is a smooth projective variety $X$ defined over a field $k$ such that after base change to an algebraic closure $\bar{k}$ it becomes isomorphic to a projective space. For example, a conic (i.e. a smooth projective curve in $\mathbb{P}^2$ of degree 2) is the main of example of a Severi-Brauer variety of dimension 1.

And finally from the point of view of Galois theory we will be interested in second Galois cohomology, i.e. the cohomology of the absolute Galois group of the base field $k$. Note that when studying this, one forgets the field $k$ itself and working just with the absolute Galois group.

It will be our goal to understand that there is a one-to-one (if properly explained) correspondence between objects defined above. This opens up a possibility of using methods of one area to the other: for example, of understanding conics via quaternion algebras, or of using cohomological techniques for a better understanding of algebraic geometry of certain varieties.

Despite broad scope of the seminar, it should be accessible for students who have basic knowledge of algebra, Galois theory and some acquaintance with algebraic geometry. At least in the beginning of the seminar we will follow the book by Gille and Szamuely (see the list of the references, so it is possible) Moreover, the schedule could be slightly adapted along the way depending on the prerequisites of the students.

Program

PDF

Talk 0 (26.10)	Introduction and overview
Talk 1 (02.11)	Quaternion algebras
Talk 2 (09.11)	Central simple algebras, cyclic algebras
Talk 3 (16.11)	Galois descent and Brauer group I
Talk 4 (23.11)	Cohomology of (profinite) groups and infinite Galois theory
Talk 5 (30.11)	Brauer group II
Talk 6 (07.12)	Recap of algebraic geometry
Talk 7 (14.12)	Severi-Brauer varieties
Talk 8 (11.01)	Severi-Brauer varieties II
Talk 9 (18.01)	Cohomological dimension and C1-fields
Talk 10 (25.01)	Residue maps for Brauer groups
Talk 11 (01.02)	Unramified Brauer group of a smooth variety and rationality

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