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Brauer group and Severi-Brauer varieties SemesterWiSe 2022 / 23 Lecturer Pavel Sechin Type of course (Veranstaltungsart) Seminar German title Brauergruppe und Severi-Brauer Varietäten Contents The seminar is dedicated to a topic that connects Galois theory, simple (non-commutative) algebras and algebraic geometry. A central simple algebra over a field 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 real numbers 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 k. A Severi-Brauer variety over k is a smooth projective variety X such that after base change to an algebraic closure of k it becomes isomorphic to a projective space. For example, a conic (i.e. a smooth projective curve of degree 2 in the projective plane) 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 works 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). Moreover, the schedule could be slightly adapted along the way depending on the prerequisites of the students. Literature
Recommended previous knowledge Galois theory, basic commutative algebra and acquaintance with algebraic geometry Time/Date Wed 12-14 Location M101 Course homepage http://homepages.uni-regensburg.de/~sep03286/Brauer.html (Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern) Registration
Preliminary meeting on 27.07 at 16.00, meet me near M229. Modules BSem, MV, MSem ECTS 4,5 |