G-torsors over affine curves - Philippe Gille скачать в хорошем качестве

G-torsors over affine curves - Philippe Gille

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/CourseNot... 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''. For more information, visit https://pcmi.ias.edu