Sha-rigidity of Chevalley groups over local rings

Abstract

We prove that every locally inner endomorphism of a Chevalley group (or its elementary subgroup) over a local ring with an irreducible root system of rank >1 (with 1/2 for the systems A_2, F_4, B_l, C_l and with 1/3 for the system G_2) is inner, so that all these groups are Sha-rigid.

14 pages