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类别的角度来看，精确类别和方案的高级格伦迪克·威特小组的系统发展，以及某些具有二元性的沃尔德豪森类别。特别是，PI将对如何消除“ 2可逆”的无处不在的假设。第二部分中，关于衍生典型结构的已知结果的财富将应用于代数理论，较高的Grethendieck-Witt理论，稳定的WITT WITT-WITT WEATT-WEALT-WEATT-WEALTERY和CYCCYCCLIC）。在第三部分中，将研究Hermitian K理论和A^1同质理论之间的关系。代理K理论，较高的Grothendieck-Witt理论，稳定的Witt理论和环状同源性是“（共同）的同源性理论”（共同）的理论“使用了TOSTUDY TOSTUDY TOSTUDY TOSTUDY tostudy TOSTUDY SYSES of POLYNOMIAL方程系统的解决方案。同一个理论首先是由代数拓扑师开发的，目的是研究在小规模下不变的几何对象的研究。后来，为了研究多项式方程的系统（在小变形下的性质可能会发生巨大变化），代数几何学家/拓扑师在代数环境中开发了类似的同谋理论。它们使我们能够利用三维空间中的直觉以及在较高维度和其他数字系统中使用多项式方程的经验（例如，在加密系统中使用），其中1+1可以等于0。在这些阶段中可以研究这些共同体学理论。但是，通常，人们可以在派生类别的级别上观察到这种“打破upinto更简单的构建块”，该类别的级别是用多项式序列系统附加的代数类别对象。该项目调查了上述派生典观与共同论理论之间的关系。