Automorphisms of Chevalley groups over commutative rings

TL;DR

This paper proves that automorphisms of Chevalley groups over commutative rings with certain invertibility conditions are all standard, combining ring, inner, central, and graph automorphisms, thus completing their classification.

Contribution

It establishes that all automorphisms of Chevalley groups over specified rings are standard, finalizing their automorphism classification.

Findings

Automorphisms are standard under given conditions.

Complete description of automorphisms of Chevalley groups.

Provides model-theoretic applications of the classification.

Abstract

In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank >1 over a commutative ring (with 1/2 for the systems A_2, F_4, B_l, C_l; with 1/2 and 1/3 for the system G_2) is standard, i.e., it is a composition of ring, inner, central and graph automorphisms. This result finalizes description of automorphisms of Chevalley groups. However the restrictions on invertible elements can be a topic of further considerations. We provide also some model-theoretic applications of this description.