Higher Grothendieck-Witt groups

高等格洛腾迪克-维特群

• 批准号: 0906290
• 负责人: Marco Schlichting
• 金额: $ 17.98万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906290&HistoricalAwards=false
• 关键词: Higher; Grothendieck; Witt; groups

The aim of this project is to deepen our understanding of the algebraic analogue of real topological K-theory: the theory of higher Grothendieck-Witt groups.The goal is to establish several fundamental results with special emphasis on eliminating restrictions on singularities and characteristics which permeate the literature. Specifically, we will study conjectures of Karoubi and Williams - the first, relating the integral homology groups of infinite orthogonal, symplectic and general linear groups, and the second relating homotopy fixed points of K-theory to hermitian K-theory. We will also study homotopy and devissage properties of higher Grothendieck-Witt groups when "2 is not invertible" which are essential in the calculation of hermitian K-groups of rings of integers in number fields. Finally, we will study higher Grothendieck-Witt groups in relation with certain invariants defined via A^1-homotopy theory.Historically, cohomology theories attach to a geometric object such as the surface of the earth (whose topological properties don't change under smalldeformations) certain algebraic objects such as a set of numbers (which are rather rigid in nature). The study of the attached algebraic objects yields information about the original geometric object. The success of cohomology theories in topology lead algebraists to define cohomology theories in an algebraic context. These algebraic cohomology theories allow us to use our intuition from 3 space and our experience with working with real numbers to study systems polynomial equations in higher dimensions and in number systems (used e.g. in cryptography) where 1+1 could be 0. The theory investigated in this project, the theory of higher Grothendieck-Witt groups, is one such algebraic cohomology theory. Compared to its companion theories - algebraic K-theory, Witt-groups and L-groups - this theory is rather underdeveloped. For instance, virtually nothing is known in relation with number systems in which1+1 = 0. This project aims to close the gap in knowledge between the 1+theory ofhigher Grothendieck-Witt groups and these companion theories.

该项目的目的是加深我们对实拓扑 K 理论的代数类比：高级 Grothendieck-Witt 群的理论的理解。目标是建立几个基本结果，特别强调消除文献中对奇点和特征的限制。具体来说，我们将研究卡鲁比和威廉姆斯的猜想 - 第一个猜想将无限正交群、辛群和一般线性群的积分同调群联系起来，第二个猜想将 K 理论的同伦不动点与厄米 K 理论联系起来。我们还将研究当“2 不可逆”时更高的 Grothendieck-Witt 群的同伦和设计性质，这对于数域中整数环的 Hermitian K 群的计算至关重要。最后，我们将研究与通过 A^1 同伦理论定义的某些不变量相关的更高的 Grothendieck-Witt 群。历史上，上同调理论附加到诸如地球表面之类的几何对象（其拓扑性质在小变形下不会改变）以及某些代数对象，例如一组数字（本质上相当刚性）。对附加代数对象的研究产生了有关原始几何对象的信息。上同调理论在拓扑学中的成功促使代数学家在代数背景下定义上同调理论。这些代数上同调理论使我们能够利用 3 空间的直觉和处理实数的经验来研究更高维度和数字系统（例如在密码学中使用）中的系统多项式方程，其中 1+1 可能为 0。本项目研究的理论，即更高的 Grothendieck-Witt 群理论，就是这样一种代数上同调理论。与它的同伴理论——代数 K 理论、Witt 群和 L 群相比——这个理论还相当不发达。例如，对于 1+1 = 0 的数字系统，我们几乎一无所知。该项目旨在缩小高级格罗腾迪克-维特群的 1+理论与这些同伴理论之间的知识差距。