Calculations in higher algebraic K-theory and related functors via derived categories

通过派生类别进行高等代数 K 理论和相关函子的计算

• 批准号: 0604583
• 负责人: Marco Schlichting
• 金额: $ 9.77万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604583&HistoricalAwards=false
• 关键词: Calculations; higher; algebraic; theory; related

The main part of the proposal deals with the systematic development ofhigher Grothendieck-Witt groups, alias hermitian K-theory, of exactcategories and schemes, from the point of view of derived categories, andof certain Waldhausen categories with duality. In particular, the PI willinvestigate how to remove the ubiquitous assumption of "2 being invertible".In a second part, the wealth of known results on the structure of derivedcategories will be applied to yield new calculations in algebraicK-theory, higher Grothendieck-Witt theory, stabilized Witt-theory andcyclic homology. In a third part, the relation between hermitian K-theoryand A^1 homotopy theory will be investigated.Algebraic K-theory, higher Grothendieck-Witt theory, stabilizedWitt-theory and cyclic homology are "(co-) homology theories" used tostudy solutions of systems of polynomial equations. Cohomology theorieshave first been developed by algebraic topologists in order to studyproperties of geometric objects which don't change under smalldeformations. Later, in order to study systems of polynomial equations(whose properties can drastically change under small deformations),algebraic geometers/topologists developed analogous cohomology theoriesin an algebraic context. They allow us to use our intuition from 3dimensional space and our experience with working with real numbers, tounderstand polynomial equations in higher dimensions, and in other numbersystems, (used e.g in cryptography) where 1+1 could be equal to 0. Inorder to study these cohomology theories one needs to break up theirvalues into simpler building blocks, which, in general, is a verydifficult problem. Frequently, however, one can observe this "breaking upinto simpler building blocks" on the level of derived categories, whichare algebro-categorical objects attached with systems of polynomialequations. This project investigates the relationship between derivedcategories and the cohomology theories mentioned above.

该建议的主要部分涉及系统的发展更高的Grothendieck-Witt群，别名埃尔米特K理论，的exactcategories和计划，从角度来看，派生类别，并与对偶的某些Waldhausen类别。在第二部分中，我们将利用已有的关于导出范畴结构的大量结果，在代数K-理论、高阶Grothendieck-Witt理论、稳定Witt理论和循环同调等方面进行新的计算。第三部分研究了Hermitian K-理论与A^1同伦理论之间的关系，代数K-理论、高阶Grothendieck-Witt理论、稳定Witt理论和循环同调是研究多项式方程组解的“（余）同调理论”。上同调理论最初是由代数拓扑学家为了研究几何对象在小变形下不发生变化的性质而发展起来的。后来，为了研究多项式方程组（其性质可以在小变形下急剧变化），代数几何学家/拓扑学家在代数背景下发展了类似的上同调理论。它们允许我们使用我们的直觉从三维空间和我们的经验与工作与真实的数字，以了解多项式方程在更高的维度，并在其他numbersystem，（用于例如在密码学）其中1+1可以等于0。为了研究这些上同调理论，人们需要把它们的值分解成更简单的积木，这通常是一个非常困难的问题。然而，人们经常可以在派生范畴的层次上观察到这种“分解成更简单的积木”，派生范畴是与多项式方程系统相连的代数范畴对象。本项目研究了导出范畴与上述上同调理论之间的关系。