Sha-rigidity of adjoint Chevalley groups of types $A_1$, $A_2$, $B_2$, $G_2$ over commutative rings

TL;DR

The paper proves Sha-rigidity of certain adjoint Chevalley groups over commutative rings by showing all locally inner endomorphisms are inner, without relying on classification or structural automorphism results.

Contribution

It establishes Sha-rigidity for specific Chevalley groups over rings, using direct proofs that do not depend on automorphism classification.

Findings

Every locally inner endomorphism is inner for these groups.

These groups are Sha-rigid over the specified rings.

Proofs are direct and do not rely on classification of automorphisms.

Abstract

We prove that every locally inner (class-preserving) endomorphism of adjoint Chevalley groups and their elementary subgroups over commutative rings is inner for the root systems A1, A2, B2 (assuming 2 is invertible in the ring), and for G2 (assuming 2 and 3 are invertible). As a consequence, these groups are Sha-rigid. The proofs are direct and do not rely on classification of automorphisms or structural results about injective endomorphisms.