Azumaya algebras with involution and classical semisimple group schemes

TL;DR

This paper establishes a classification of classical semisimple group schemes over schemes with 2 invertible via Azumaya algebras with involution, extending known results from fields to more general base schemes.

Contribution

It constructs an equivalence between Azumaya algebras with involution and classical semisimple group schemes over schemes, generalizing the classification from fields to schemes.

Findings

Classification of adjoint group schemes via Azumaya algebras with involution

Application to specialization of Azumaya algebras over Henselian rings

Implications for the Grothendieck-Serre conjecture

Abstract

Let $S$ be a non-empty scheme with 2 invertible. In this paper we present a functor $F: AZ_*^{n'} \rightarrow GS_*^n$ where $AZ_*^{n'}$ and $GS_*^n$ are fibered categories over $Sch_S$ given respectively by degree-$n'$ Azumaya algebras with an involution of type $*$ and rank-$n$ adjoint group schemes of classical type $*$ with absolutely simple fibers. Here $n'$ is a function of $n$. We show that this functor is an equivalence of fibered categories using \'etale descent, thus giving a classification of adjoint (as well as simply connected) group schemes over $S$, generalizing the well known case when the base scheme is the spectrum of a field. In particular, this implies that every adjoint group scheme of classical type with absolutely simple fibers is isomorphic to the neutral component of the automorphism group scheme of a unique (up to isomorphism) Azumaya algebra with involution. We also show interesting applications of this classification such as specialization theorem for isomorphism classes of Azumaya algebra with involution over Henselian local rings, uniqueness of integral model for groups with good reduction over discrete valued fields and discuss its implications on the Grothendieck-Serre conjecture over certain domains.