New Developments in Algebraic K-Theory

“代数K-理论的新发展”项目摘要这个项目涉及到域的代数K-理论。在过去的两年里，V.Voevodsky用他在证明Milnor猜想时使用的方法彻底改变了代数K理论。他开发的方法表明，将在未来几年开发一个程序，通过所谓的代数簇的稳定同伦范畴来理解代数K-理论。使用他的方法，可以在某些情况下验证Quillen Lichtenbaum猜想，特别是对于素数为2的数域。这个项目的一部分是继续这项工作，一些与M.Rost合作，在更多的情况下证明这一猜想。G.Carlsson发展了Quillen Lichtenbaum猜想的一个修正版本，它应该显式地确定代数K-理论谱的所有同伦群，而不仅仅是像原始猜想中那样的高维同伦群。这个版本涉及绝对Galois群的表示理论，而不仅仅是Galoios上同调。这个项目的第二个目标是了解这两个结构之间的关系，希望Voevodsky的方法可以证明修订后的猜想，或者Carlsson的方法将阐明Voevodsky的工作和Quilen Lichtenbaum猜想之间的一般关系。代数K-理论在数学中是一个相对较新的学科，它承诺将两个看起来非常不同的领域联系起来，即拓扑学(研究曲线和曲面及其变形)和非常经典的数论领域，它研究整数和整数上的方程的性质。在拓扑学上定义的代数K-理论似乎携带了大量的算术信息。关于这一关系的两个重要猜想是Quillen Lichtenbaum猜想和Bloch-Kato猜想。在过去的两三年里，V.Voevodsky在这两个猜想上的突破性工作彻底改变了这一领域。这个项目将致力于进一步发展这项工作，让Voevodsky和Rost继续他们关于Bloch-Kato猜想的工作，并通过将Voevodsky的方法与Carlsson提出的Quillen Lichtenbaum猜想的修正的、更强的版本联系起来。从长远来看，这些问题的解有可能帮助我们研究代数方程的整数解集，例如著名的费马方程。

Project Abstract for "New Developments in Algebraic K-Theory"This project concerns the algebraic K-theory of fields. In the last two years, V. Voevodsky has revolutionized algebraic K-theory with the methods he uses in his proof of the Milnor conjecture. The methods he has developed suggest a program which will be developed over the next few years to understand algebraic K-theory through a so-called stable homotopy category of algebraic varieties. Using his methods, it has been possible to verify the Quillen Lichtenbaum conjecture in some cases, notably for number fields at the prime two. Part of this project is to continue this work, some joint with M. Rost, to prove this conjecture in a larger number of cases. G. Carlsson has developed a revised version of the Quillen Lichtenbaum conjecture, which should explicitly identify all homotopy groups of the algebraic K-theory spectra, not only the high dimensional ones as in the original conjecture. This version involves the representation theory of the absolute Galois group, rather than just the Galoios cohomology. The second goal of this project is to understand the relationship between The two constructions, in the hopes that Voevodsky's method might prove the revised conjecture, or that Carlsson's method will shed light on the general relationship between Voevodsky's work and the Quilen Lichtenbaum conjecture.Algebraic K-theory is a relatively new subject in mathematics, which promises to relate two seemingly very disparate areas, namely topology (the study of curves and surfaces and their deformations), and the very classical area of number theory, which studies the properties of the integers and equations over the integers. Algebraic K-theory, which is defined topologically, appears to carry with it a great deal of arithmetic information. Two important conjectures concerning this relationship are the Quillen Lichtenbaum conjecture and the Bloch-Kato conjecture. V. Voevodsky has revolutionized this area over the last two or three years with his breakthrough work on both conjectures. This project will be devoted to developing this work further, by having Voevodsky and Rost continue their work on the Bloch-Kato conjecture, and by connecting Voevodsky's methods with a revised, stronger version of the Quillen Lichtenbaum conjecture proposed by Carlsson. In the long run, solutions to these problems have the potential to help us study the sets of solutions in the integers of algebraic equations, such as the famous Fermat equation.