VERSAL TORSORS AND RETRACTS

Abstract

Let G be an algebraic group over F and p a prime integer. We introduce the notion of a p-retract rational variety and prove that if Y → X is a p-versal G-torsor, then BG is a stable p-retract of X. It follows that the classifying space BG is p-retract rational if and only if there is a p-versal G-torsor Y → X with X a rational variety, that is, all G-torsors over infinite fields are rationally parameterized. In particular, for such groups G the unramified Galois cohomology group $$ {H}_{\mathrm{nr}}^n $$ (F(BG), ℚp/ℤp(j)) coincides with Hn(F, ℚp/ℤp(j)).