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

Louisiana State University, Baton Rouge LA

Investigators

Abstract

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