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

除环是一个非常基本的对象。它只是一个具有结合乘积结构的向量空间，其中每个非零元素都有一个逆。经典的Wedderburn定理表明除环是环论的基本组成部分。实际上，从一个具有某种温和有限条件的任意环R出发，R/J（R）可以表示为除环上矩阵的乘积。研究除代数的K-理论的方法之一（它直接应用于这些对象的群结构）是考虑约化范数映射的核。这个群被称为约化怀特黑德群，SK 1，对它的研究是约化K理论的主题（更多细节请参见项目描述）。虽然最早的专门研究约化Whitehead群的工作是由Tannaka和Nakayama在20世纪40年代中期发表的，（证明了局部域上的除环的SK 1是平凡的）和20世纪50年代由Artin的博士生Wang（整体域上的除环的SK 1是平凡的），1976年，V. Platonov向前迈出了一大步，他构建了这个群体并非微不足道的例子。这回答了几个问题所提出的名人，如山雀，Keneser，塞尔和博雷尔在设置的代数群的负面，从而开辟了新的研究路线，在部门代数理论和代数群。尽管过去了半个多世纪，仍然有很大的兴趣在减少怀特海集团SK 1，和充满活力的活动，围绕这一主题主要是由于新的技术，从估值理论和代数几何。申请人与Adrian沃兹沃斯的最近工作，通过在分次除代数的设置中引入约化的Whitehead群，并将计算的复杂性转移到该设置而不是直接使用给定的除代数，不仅为该组带来了新的启示，而且可以以更容易理解的系统方式重新获得文献中获得的大部分结果。如果人们试图遵循先前为获得这一主题的结果而采用的论点和方法，这一点将是清楚的。这一点在幺正情况下更加明显：即使在过去了大约30年之后，在幺正情况下计算SK 1似乎没有任何改进，直到我们的工作出现。

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.