Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Azumaya algebras with involution and classical semisimple group schemes

S. Srimathy

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 étale 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.

Azumaya algebras with involution and classical semisimple group schemes

Abstract

Let be a non-empty scheme with 2 invertible. In this paper we present a functor where and are fibered categories over given respectively by degree- Azumaya algebras with an involution of type and rank- adjoint group schemes of classical type with absolutely simple fibers. Here is a function of . We show that this functor is an equivalence of fibered categories using étale descent, thus giving a classification of adjoint (as well as simply connected) group schemes over , 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.

Paper Structure

This paper contains 9 sections, 13 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Group schemes over an arbitrary scheme
  4. Azumaya algebras with involution over an arbitrary scheme
  5. The Group scheme of Automorphisms of Azumaya algebras with involution
  6. The fibered categories $Az_*^n$ and $GS_*^n$
  7. Equivalence of $Az_*^{n'}$ and $GS_*^n$
  8. Applications
  9. Acknowledgements

Key Result

Proposition 3.3

Let $G$ and $H$ be adjoint group schemes over $S$ with absolutely simple fibers of a given type and rank. Then locally for the étale topology on $S$, $G$ and $H$ are isomorphic. In fact, any reductive group scheme with root datum $\mathcal{R}$ is étale locally isomorphic to a unique Chevalley group

Theorems & Definitions (41)

  • Remark 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • proof
  • Remark 3.7
  • Definition 4.1
  • ...and 31 more