Applications of the motivic Becker-Gottlieb transfer

奖项：DMS 1200284，首席研究员：罗伊·约书亚在过去的三年里，PI与斯坦福大学的Gunnar Carlsson一起参与了一项研究计划，目标是构建和建立Becker-Gottlieb转移的动机变体，并用它来研究有限系数动机上同调和代数K-理论中的几个问题。过去三年的这种合作工作已经产生了两份预印本，在那里已经建立了这样一种转移，并为随后的工作建立了一个基本框架。本研究方案的目的是探索这种转换在有限系数上同调和代数K-理论中的应用，并预期有几个这样的应用。代数拓扑学中相应的经典概念在解决亚当斯？猜想，其中转移的使用提供了一个复杂版本的分裂原则。所提出的研究的一个主要主题是，相同原理的其他更复杂的应用解释了代数K-理论中的几个更深层次的结果：例如，将Thomason关于底元倒置到Hilbert定理90的Mod-L代数K-理论的上同调下降的著名结果的约化，以及将mod-L代数K-理论的刚性性质约化到Picard群。然而，在没有类似Becker-Gottlieb转会的情况下，这些削减是相当漫长和困难的，需要开发定制的精密机械。目前也有一些类似的技巧似乎可以应用于一些公开的问题：例如，Carlsson的猜想将代数K-理论与某些与域的Galois群相关的等变K-群联系起来，还有几个关于代数群的分类空间的Motivic上同调的公开问题。Motivic上同调和Motivic上同伦理论是从代数拓扑到代数几何的成熟技术的相对较新的应用。传统上，上同调和同伦方法被用来区分拓扑空间和分析它们的性质。另一方面，代数几何处理作为多项式方程的零点出现的特殊空间。直到最近几年，在使用同伦方法解决了一个名为Milnor猜想的公开问题后，这种技术在代数几何中的相关性和实用性才浮出水面。考虑到Becker-Gottlieb变换是代数拓扑学中最通用的工具之一，在Motivic上下文中使用其变体是合理的。这个项目的重要性和相关性源于这些观察。

AbstractAward: DMS 1200284, Principal Investigator: Roy JoshuaThe PI, together with Gunnar Carlsson at Stanford, has been involved in a research program over the last three years with the goal of constructing and setting up a motivic variant of the Becker-Gottlieb transfer and using this to study several problems in motivic cohomology and algebraic K-theory with finite coefficients. This collaborative work over the last three years has already resulted in two preprints, where such a transfer has already been constructed and a basic framework for subsequent work established. The goal of the present research proposal is to explore applications of this transfer to motivic cohomology and algebraic K-theory with finite coefficients with several such applications expected. The corresponding classical notions in algebraic topology have played a key role in settling important problems like the Adams? conjecture, where the use of the transfer provided a sophisticated version of the splitting principle. A leading theme of the proposed research is that other more sophisticated applications of the same principle accounts for several of the deeper results in algebraic K-theory: for example, the reduction of Thomason's well-known result on cohomological descent for mod-l algebraic K-theory with the Bott-element inverted to Hilbert's theorem 90, and the reduction of rigidity properties of mod-l algebraic K-theory to Picard groups. However, in the absence of an analogue of the Becker-Gottlieb transfer, these reductions have been rather long and difficult, requiring the development of customized elaborate machinery. There are also currently several open problems to which similar techniques seem to apply: for example, Carlsson's conjectures relating the algebraic K-theory of fields with certain equivariant K-groups associated to the Galois group of the field and also several open questions on the motivic cohomology of the classifying spaces of algebraic groups.Motivic cohomology and motivic homotopy theory are relatively new applications of well-established techniques from algebraic topology to algebraic geometry. Classically cohomology and homotopy methods are used to distinguish between topological spaces and to analyze their properties. Algebraic geometry, on the other hand, deals with special spaces arising as zeroes of polynomial equations. It is only in recent years, after the solution of an open problem called the Milnor conjecture using homotopy methods that the relevance and usefulness of such techniques in algebraic geometry have come to light. Considering that the Becker-Gottlieb transfer is one of the most versatile tools in algebraic topology, it is reasonable to expect much progress using a variant of it in the motivic context. The importance and relevance of this project stems from these observations.