Homology of Linear Groups with Applications to Algebraic K-theory

线性群的同调及其在代数 K 理论中的应用

• 批准号: 0242906
• 负责人: Kevin Knudson
• 金额: $ 1.91万
• 依托单位: Mississippi State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-15 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0242906&HistoricalAwards=false
• 关键词: Homology; Linear; Groups; Applications; Algebraic

Knudson studies the homology of linear groups over various rings of interest in algebraic K-theory. In this project he focuses attention on the low-dimensional homology groups of the general linear group over the coordinate ring of an elliptic curve over a number field with the goal of proving that the second K-group of such a curve has finite rank. This extends Knudson's previous work on the homology of rank one groups over such rings. In addition, a new homology theory based on algebraic cycles in the product of a scheme with the simplicial classifying scheme of an algebraic group is constructed and its properties investigated. The hope is that this construction will lead to a proof of the Friedlander-Milnor conjecture concerning the homology of algebraic groups made discrete. Also, the investigator's previous work on the structure of special linear groups over integral Laurent polynomial rings is used in the study of the Burau representation of the braid groups. This representation is known to be faithful for three strings and unfaithful for five or more strings; the remaining case of four strings is singled out for study in this project. Finally, Knudson studies the completion of a discrete group relative to a Zariski dense representation in a reductive group over a field of positive characteristic. This generalizes the classical unipotent completion of a group and extends to positive characteristic R. Hain's work in characteristic zero.A scheme is a geometric object constructed from solution sets of polynomial equations. Algebraic K-theory associates to a scheme a sequence of groups which encode information about the scheme. One aspect of this project is the study of the K-groups of an elliptic curve. Such curves have remarkably rich structure and appear in various branches of mathematics such as algebraic geometry and coding theory. A seemingly unrelated part of this project concerns the so-called Burau representation of the braid groups which is intimately connected with knot theory. These diverse topics are unified by studying the structure of groups of matrices with entries in various rings (the coordinate ring of the elliptic curve in the first case and the ring of integral Laurent polynomials in the second). The hope is to solve several outstanding conjectures about these objects.

Knudson studies the homology of linear groups over various rings of interest in algebraic K-theory. 在这个项目中，他将注意力集中在数域上椭圆曲线坐标环上的一般线性群的低维同调群上，目的是证明该曲线的第二个 K 群具有有限秩。 This extends Knudson's previous work on the homology of rank one groups over such rings. 此外，还构建了一种新的基于代数环的同调理论，并研究了其性质。 希望这种构造能够证明​​弗里德兰德-米尔诺猜想，该猜想涉及离散代数群的同源性。 此外，研究者之前关于积分洛朗多项式环上的特殊线性群的结构的工作也被用于辫子群的 Burau 表示的研究。 This representation is known to be faithful for three strings and unfaithful for five or more strings; the remaining case of four strings is singled out for study in this project. 最后，Knudson 研究了在正特征域上，离散群相对于还原群中 Zariski 稠密表示的完成性。 这概括了群的经典单能完成，并扩展到正特征 R。Hain 在特征零方面的工作。方案是由多项式方程的解集构造的几何对象。 Algebraic K-theory associates to a scheme a sequence of groups which encode information about the scheme. One aspect of this project is the study of the K-groups of an elliptic curve. Such curves have remarkably rich structure and appear in various branches of mathematics such as algebraic geometry and coding theory. 该项目的一个看似无关的部分涉及所谓的辫子群的 Burau 表示，它与结理论密切相关。 这些不同的主题通过研究具有不同环（第一种情况下的椭圆曲线的坐标环和第二种情况下的积分洛朗多项式的环）中的条目的矩阵组的结构来统一。 The hope is to solve several outstanding conjectures about these objects.