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.

该提案的主要部分涉及从派生范畴的角度以及某些具有对偶性的 Waldhausen 范畴的系统发展高级 Grothendieck-Witt 群（又名 Hermitian K 理论）的精确范畴和方案。特别是，PI 将研究如何消除普遍存在的“2 可逆”假设。在第二部分中，派生类别结构上的大量已知结果将应用于代数 K 理论、高等 Grothendieck-Witt 中的新计算理论、稳定维特理论和循环同调。第三部分将研究厄米K-理论和A^1同伦论之间的关系。代数K-理论、高等Grothendieck-Witt理论、稳定维特理论和循环同调都是用于研究解的“（共）同调理论”多项式方程组。上同调理论最初是由代数拓扑学家提出的，目的是研究在小变形下不发生变化的几何对象的性质。后来，为了研究多项式方程组（其性质在小变形下会发生巨大变化），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。它们使我们能够利用 3 维空间的直觉和处理实数的经验，理解更高维度的多项式方程以及其他数字系统（例如在密码学中使用），其中 1+1 可能等于 0。这些上同调理论需要将它们的值分解为更简单的构建块，这通常是一个非常困难的问题。然而，人们经常可以在派生类别的层面上观察到这种“分解为更简单的构建块”，派生类别是附加于多项式方程系统的代数分类对象。该项目研究派生范畴与上述上同调理论之间的关系。