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

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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.