Birational geometry and arithmetic of linear algebraic groups. I (Q2718156)

This is the first part of a long survey concerning the geometry and the arithmetic of linear algebraic groups. This first part has two chapters. In the first chapter the author presents some general facts from scheme theory and cohomology (such as group schemes and their cohomology, the Brauer group of projective varieties and \(k\) forms) that will be used in the sequel. Here are the main topics discussed: group objects in categories, group schemes, affine groups and their Hopf algebras, group schemes over fields, algebraic groups, diagonal groups, Galois cohomology, étale sheaves and étale cohomology, Weil and Cartier divisors, general facts on the Brauer group of a projective variety, the Kummer exact sequence, Tate groups, Picard and Lefschetz numbers, forms and \(1\)-dimensional cohomology, the decomposition field of a form, forms of group schemes, the Brauer group of a field, the theorem of Chevalley, semisimple groups, etc.NEWLINENEWLINENEWLINEIn the second chapter the author studies the birational geometry of algebraic tori: the variety of maximal tori of a reductive group, the structure of general tori of semisimple groups, the Picard and the Brauer group of linear algebraic groups, projective models of linear algebraic groups, tori with cyclic decomposition field and their birational classification, stably-rational varieties (e.g. stably-rational tori). This second chapter contains also some new results.NEWLINENEWLINENEWLINEFor part III and IV of this paper see: Vestn. Samar. Gos. Univ., Mat. Mekh. Fiz. Khim. Biol. 1998, No. 2(8), 5--54 (1998; Zbl 1038.14019) and 1999, No. 2(12), 5--47 (1999; Zbl 1034.14006).