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

点击翻译按钮获取中文摘要

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.