Sha-rigidity of Chevalley groups over local rings

TL;DR

This paper proves that Chevalley groups over local rings with certain root systems are Sha-rigid, meaning all locally inner endomorphisms are actually inner, confirming their structural rigidity.

Contribution

It establishes Sha-rigidity for Chevalley groups over local rings with specific root systems, extending understanding of their automorphism structures.

Findings

All locally inner endomorphisms are inner for these groups.

Chevalley groups over such rings are Sha-rigid.

Results depend on the root system's rank and specific divisibility conditions.

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.