Flasque resolutions of linear reductive group schemes

Abstract

This paper studies flasque resolutions for linear reductive group schemes over arbitrary bases, extending Colliot-Thelene's theory. A flasque resolution exists for any linear reductive group scheme over a connected base scheme (resp. a quasi-compact, quasi-separated base scheme). The construction relies on Borovoi's fundamental group and descent theory involving automorphisms of Chevalley groups. Applications include R-equivalence for reductive group schemes over rings and over a field of cohomological dimension at most 2.