Graded approach to the theory of division algebras, with applications to reduced K-theory

除法代数理论的分级方法及其在简化 K 理论中的应用

• 批准号: EP/I007784/1
• 负责人: Roozbeh Hazrat
• 金额: $ 4.17万
• 依托单位: Queen's University Belfast
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2011
• 资助国家: 英国
• 起止时间: 2011 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FI007784%2F1
• 关键词: Graded; approach; theory; division; algebras

A division ring is a very elementary object. It is just a vector space with an associative product structure where each non- zero element has an inverse. The classical theorem of Wedderburn shows that division rings are basic building blocks of ring theory. Indeed starting from an arbitrary ring R with some mild finite condition, R/J(R) can be expressed as a product of matrices over division rings. One of the approaches for studying the K-theory of division algebras (which has direct application into the group structure of these objects) is to consider the kernel of the reduced norm map. This group is called the reduced Whitehead group, SK1, and the study of it is the subject of reduced K-theory (see the description of the project for more detail). Although the earliest work specifically on the reduced Whitehead group was published in mid 1940's by Tannaka and Nakayama (proving that SK1 of division rings over local fields are trivial) and in the 1950's by Artin's PhD student Wang (SK1 of division rings over global fields are trivial), a giant step forward was taken by V. Platonov in 1976 who constructed examples that this group is non-trivial. This answered several questions raised by luminaries such as Tits, Keneser, Serre and Borel in the setting of algebraic groups in negative and thus opened up new lines of research in division algebra theory and algebraic groups. Despite the passing of more than half a century, there is still a substantial interest in the reduced Whitehead group SK1, and vibrant activities around this subject mostly thanks to the new techniques from valuation theory and algebraic geometry. The recent work of the applicant with Adrian Wadsworth, by introducing the reduced Whitehead group in the setting of graded division algebras, and carrying over the complexity of calculations to this setting instead of directly working with a given division algebra, has not only shed new light into this group, but could recapture most of the results obtained in the literature in a systematic way which is much easier to follow. This would be clear if one tries to follow the arguments and methods employed previously to obtain results in this subject. This is even more apparent in the unitary case: even after the passage of some 30 years, there does not seem to have been any improvement in calculating SK1 in the unitary setting until the appearance of our work.

除法环是一个非常基本的物体。它只是一个具有结合积结构的向量空间，其中每个非零元素都有一个逆。经典的Wedderburn定理表明除环是环理论的基本组成部分。实际上，从具有温和有限条件的任意环R出发，R/J(R)可以表示为除法环上矩阵的乘积。研究除法代数的k理论（直接应用于这些对象的群结构）的方法之一是考虑约简范数映射的核。这个群体被称为简化Whitehead群，SK1，对它的研究是简化k理论的主题（详见项目描述）。虽然最早的关于约简Whitehead群的工作是在1940年代中期由Tannaka和Nakayama发表的（证明局部域上的除法环的SK1是平凡的），以及在1950年代由Artin的博士生Wang发表的（全局域上的除法环的SK1是平凡的），但V. Platonov在1976年迈出了巨大的一步，他构造了这个群是非平凡的例子。这回答了Tits、Keneser、Serre、Borel等著名学者在负代数群的设置中提出的几个问题，从而开辟了除法代数理论和代数群研究的新方向。尽管半个多世纪过去了，人们对缩减的怀特黑德群SK1仍然有很大的兴趣，围绕这一主题的活跃活动主要归功于估值理论和代数几何的新技术。申请人与Adrian Wadsworth最近的工作，通过在分级除法代数的设置中引入简化Whitehead群，并将计算的复杂性转移到该设置中，而不是直接使用给定的除法代数，不仅为该组提供了新的思路，而且可以以系统的方式重新获得文献中获得的大多数结果，这更容易遵循。如果一个人试图遵循先前在这一主题中用于获得结果的参数和方法，这一点就会很清楚。这在单一的情况下更加明显：即使在大约30年的过去之后，在单一设置中计算SK1似乎没有任何改进，直到我们的工作出现。