论文标题
Azumaya代数，具有不合适性和经典的半神经群计划
Azumaya algebras with involution and classical semisimple group schemes
论文作者
论文摘要
令$ s $为具有2个可逆的非空计划。在本文中，我们介绍一个函子$ f：az _*^{n'} \ rightarrow gs _*^n $其中$ az _*^{n'} $和$ gs _*^n $分别由$ sch_s $ ival tem-sch_s $ fible-n'u $ n'u $ azumaya $ n schem schem schem schem schem schem schem和schem schem schem schem schem schem schem schem schem schem an $*$，带有绝对简单的纤维。这里$ n'$是$ n $的函数。我们表明，该函子是使用étale下降的纤维类别的等效性，因此在$ s $上进行了伴随（以及简单地连接）组方案的分类，当基本方案是磁场的频谱时，概括了众所周知的情况。特别是，这意味着，具有绝对简单纤维的经典类型的每个伴随组方案都是与独特的（直至同构）Azumaya代数的自动形态组方案的中性成分同构。我们还展示了此分类的有趣应用，例如针对Azumaya代数的同构类别的专业定理，对Henselian本地环的不合格，对于离散有价值领域的良好降低的组的积分模型的独特性，并讨论了其对Grothendieck-Serre的含义，对Grothendieck-Serre的含义。
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.
