R-equivalence and retract rationality of reductive groups

by Anastasia Stavrova (SPbU)

Thursday 18 Nov 2021, 11:30 → 13:30 Europe/Moscow

818-1526-4739 (Zoom)

The $R$-equivalence is an equivalence relation on points of algebraic varieties over a field that was introduced bu Yuri Manin and proved to be very fruitful in the study of algebraic groups. In particular, it has been known that if $G$ is an algebraic torus or a simply connected isotropic semisimple reductive group over a field $k$, then $G(K)/R=1$ for all field extensions $K$ of $k$ if and only if $G$ is a retract rational $k$-variety, i.e. a retract of an open subset of an affine space (J.-L. Colliot-Thelene and J.-J. Sansuc, 1987; P. Gille, 2007). In the second case of simply connected isotropic groups, the same is moreover equivalent to the local triviality of the $0$-th $\mathbb{A}^1$-homotopy group of $G$ (A. Asok, M. Hoyois, M. Wendt, 2018). I will discuss generalizations of these facts to the case of groups over semilocal rings. The talk is based on the joint work with P. Gille arXiv:2107.01950.

Video