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

本文的主要部分从派生范畴和某些具有对偶性的Waldhausen范畴的观点出发，系统地发展了精确范畴和方案的更高Grothendieck-Witt群，即厄米特k理论。特别是，PI将研究如何消除普遍存在的“2是可逆的”假设。在第二部分中，关于衍生范畴结构的大量已知结果将被应用于代数理论、更高Grothendieck-Witt理论、稳定witt理论和循环同调中的新计算。第三部分研究了厄米特k理论与a ^1同伦理论之间的关系。代数k理论、高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“（协）同调理论”。上同调理论最初是由代数拓扑学家提出的，目的是研究几何物体在小变形下不变化的性质。后来，为了研究多项式方程系统（其性质可以在微小的变形下急剧改变），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在三维空间的直觉和我们处理实数的经验，来理解更高维度的多项式方程，以及在其他数字系统中（例如在密码学中），1+1可能等于0。为了研究这些上同理论，我们需要把它们的值分解成更简单的构建块，这通常是一个非常困难的问题。然而，人们经常可以在派生范畴的层面上观察到这种“分解成更简单的构建块”，这些范畴是与多项式方程系统相连的代数范畴对象。本课题研究了派生范畴与上同论之间的关系。

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.