On the Grothendieck-Serre Conjecture for Classical Groups

TL;DR

This paper advances the understanding of the Grothendieck-Serre conjecture for classical groups by constructing an explicit Gersten-Witt complex for Witt groups of Azumaya algebras with involution, proving new cases especially in low dimensions.

Contribution

It introduces a new explicit construction of the Gersten-Witt complex for Witt groups of Azumaya algebras with involution on regular semilocal rings, extending the conjecture's validity.

Findings

Gersten-Witt complex constructed explicitly with second residue maps

Complex shown to be exact for rings of dimension ≤ 2

Extended results to rings of dimension ≤ 4 under additional hypotheses

Abstract

We prove some new cases of the Grothendieck-Serre conjecture for classical groups. This is based on a new construction of the Gersten-Witt complex for Witt groups of Azumaya algebras with involution on regular semilocal rings, with explicit second residue maps; the complex is shown to be exact when the ring is of dimension $\le 2$ (or $\le 4$, with additional hypotheses on the algebra with involution). Note that we do not assume that the ring contains a field.