| Abstrakt: |
Let S be a non-empty scheme with 2 invertible. In this paper, we present a functor F: AZn′ ∗ → GSn∗, where AZn′ * and GSn* are fibered categories over SchS 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 a specialization theorem for isomorphism classes of Azumaya algebra with involution over Henselian local rings and the uniqueness of an integral model for groups with good reduction over discrete valued fields, and we discuss its implications for the Grothendieck-Serre conjecture over certain domains. [ABSTRACT FROM AUTHOR] |
