New methods in algebraic K-theory

代数 K 理论的新方法

• 批准号: 424239956
• 负责人: Professor Dr. Markus Land
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2019-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/424239956?language=en
• 关键词: New; methods; algebraic; theory

Algebraic K-theory studies rings through their projective modules and geometric objects like schemes through their vector bundles. It is hence closely related to questions in algebra and algebraic geometry. However, the algebraic K-theory of group rings plays an important role in algebraic topology, in the classification of manifolds: It can decide whether the two boundaries of a given h-cobordism are diffeomorphic. This is the basis of surgery theory, a method of systematically studying the classification of manifolds. It was a vision of Waldhausen, that the K-theory of spectral group rings, an object of stable homotopy theory or higher algebra, is closely related to the group of diffeomorphisms of a manifold. This vision is one of the motivations for studying the K-theory of ring spectra.In this project, one the hand one we wish to access further insight to excision phenomena in K-theory and on the other hand aim to connect a variant of algebraic K-theory to questions of geometric topology. In the first part, the aim is thus to understand structural properties of K-theory which are of particular help for concrete new calculations. The basis for the approach we wish to take in this project is a recent result of Tamme and myself, which describes the failure of excision effectively: For every excision context, we construct a map of ring spectra which determines the failure of excision. One of the two ring spectra involved is part of the excision context, the other, however, is new and constructed out of the given excision context. Many known results follow simply from the existence and formal properties of this new ring spectrum. This project is thus about getting more subtle insight into this new ring, and to use this for questions in excision and concrete calculations.In the second part of the project, we want to connect algebraic and hermitian algebraic K-theory to geometric questions. The basis here is the new construction of a genuine C_2 spectrum KR, called real algebraic K-theory, whose underlying spectrum is algebraic K-theory, whose fixed points, hermitian K-theory, are described as an algebraic cobordism category, and whose geometric fixed points are algebraic L-theory. As in Waldhausen's vision, the real algebraic K-theory of ring spectra is expected to be closely related to geometric questions. The new construction of KR allows to study the real algebraic K-theory of such ring spectra, and the goal of the second part of the proposed project is to draw conclusions of the gained knowledge in question of geometric topology.

代数K-理论通过它们的投射模研究环，通过它们的向量丛研究几何对象。因此，它与代数和代数几何中的问题密切相关。然而，群环的代数K-理论在代数拓扑学和流形的分类中起着重要的作用：它可以决定给定的h-协边的两个边界是否同构。这是外科理论的基础，是系统研究流形分类的方法。这是一个愿景的瓦尔德豪森，即K理论的谱群环，一个对象的稳定同伦理论或高等代数，是密切相关的一组同胚的一个流形。这一愿景是动机之一的研究K-理论的环spectross.In这个项目中，一方面，我们希望获得进一步的洞察力切除现象的K-理论，另一方面，旨在连接一个变种的代数K-理论的问题的几何拓扑。因此，在第一部分中，目的是理解K理论的结构性质，这些性质对具体的新计算特别有帮助。在这个项目中，我们希望采取的方法的基础是最近的结果Tamme和我自己，它有效地描述了切除的失败：对于每个切除的情况下，我们构建了一个地图的环光谱，确定切除的失败。所涉及的两个环谱之一是切除上下文的一部分，然而，另一个是新的，并构造出给定的切除上下文。许多已知的结果简单地遵循这个新的环谱的存在性和形式性质。因此，这个项目是关于获得更微妙的洞察到这个新的环，并使用它的问题在切除和具体的计算。在项目的第二部分，我们希望连接代数和埃尔米特代数K理论的几何问题。这里的基础是一个真正的C_2谱KR的新构造，称为真实的代数K-理论，它的基本谱是代数K-理论，它的不动点（埃尔米特K-理论）是用代数配边范畴来描述的，它的几何不动点是代数L-理论。在瓦尔德豪森的视野中，环谱的真实的代数K-理论预计将与几何问题密切相关。新的建设KR允许研究的真实的代数K-理论等环谱，和拟议项目的第二部分的目标是得出结论所获得的知识问题的几何拓扑结构。