Preliminary versions and articles in preparation
Caution: the following articles are preliminary versions. Most proofs are complete, but I will rewrite and reorganize them!
Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem
Abstract:
We study higher Grothendieck-Witt groups, alias algebraic hermitian K-theory, in the framework of exact categories with weak euqivalences and duality; generalizing Karoubis's hermitian K-theory of rings.
We prove a version of Thomason localization for this theory (when 2 is invertible), extending an exact sequence of Walter to the left.
As corollary, we obtain a new, algebraic, and more general proof of Karoubi's fundamental theorem in hermitian K-theory. The localization theorem also implies various (Zariski, Nisnevich) descent theorems for hermitian K and G-theory of (possible singular schemes) provided "2 is invertible".
We also explain the relation of hermitian K-theory to Ranicki's L-groups and Balmer's triangular Witt groups.
Higher algebraic K-theory (after Quillen, Thomason, and others)
Abstract:
These are the notes (to be expanded in the future) for a course taught by the author at the Sedano Winter school on K-theory, January 23-26 2007 in Sedano, Spain.
We give a short introduction (with a few proofs) to higher algebraic K-theory (mainly of schemes) based on the work of Quillen, Waldhausen, Thomason and others.
Descent properties of Witt groups
Abstract:
Throughout this article, we assume "2 to be invertible".
We show that triangular Witt-groups don't satisfy Zariski-Mayer-Vietoris for singular schemes, in general.
We prove that a modified version - stabilized Witt groups or ultimate lower L-groups - does satisfy Zariski-Mayer-Vietoris and has the same value on smooth schemes as triangular Witt-groups.
We prove furthermore that stabilized Witt-groups satisfy the Mayer-Vietoris property for elementary Nisnivich squares and, in characteristic 0, for abstract blow-up squares (for possibly singular schemes).
We also show that they are homotopy invariant.
As a corollary, we obtain that triangular Witt-groups are homotopy invariant even for singular schemes.
Matsumoto's theorem for quadratic forms
Abstract:
This is a draft of a paper which will not be published since its main result - an isomorphism between K_2^{MW}(F) and (-1)-hermitian K_2 of a field F, - is an old result of Suslin.
Marco Schlichting