G-torsors over affine curves - Philippe Gille
Channel: IAS | Park City Mathematics Institute | PCMI
Date: 13/07/2021

2021 Graduate Summer School

Topic: G-torsors over affine curves
Speaker: Philippe Gille 
Affiliation: Université Claude Bernard Lyon 1
Date: July 12, 2021


We shall present the theory of G-torsors (or G-bundles) in algebraic geometry which includes for example vector bundles and quadratic bundles (Grothendieck-Serre, 1958). We focus on the case of an affine smooth connected curve firstly over an algebraically closed field k; we shall show that G-torsors are trivial for a semisimple k-group G. Secondly we will consider the case of a perfect field and discuss the important case of the affine line (Raghunathan-Ramanathan, 1984). This will be an opportunity to deal with étale cohomology and patching techniques.

References:

V. Chernousov, P.  Gille, A. Pianzola, Three-point Lie algebras and Grothendieck's dessins d'enfants. Math. Res. Lett. 23 (2016), 81–104.

J.S. Milne,  Lectures on etale cohomology, https://www.jmilne.org/math/CourseNotes/LEC.pdf

M. S. Raghunathan, A. Ramanathan,  Principal bundles on the affine line. Proc. Indian Acad. Sci. Math. Sci. 93 (1984), 137–145.

Background:

For the background, I recommend to know basic algebraic geometry for example about flatness and smoothness and also a bit on linear algebraic groups.

For the first topic, the reference is Hartshorne (or Milne, Etale cohomology, the beginning). For the second one the reference is Milne's book "Algebraic groups''. 
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