Marco Schlichting's Research |
in dvi,
format.
Abstract: We prove a localization theorem for the hermitian K-theory of regular rings analogous to a well-known theorem in algebraic K-theory. Our proof relies among others on a study of derived categories and on a generalization of a theorem of Pedersen-Weibel to the hermitian setting. |
in dvi,
ps resp. pdf
format.
Abstract: I give an example of two closed model categories having equivalent homotopy categories but different Waldhausen $K$-theories. I also show that there cannot exist a functor from small triangulated categories to spaces which recovers Quillen's $K$-theory for exact categories and which satisfies localization. |
in dvi,
ps, resp. pdf
format and the figure.ps file (appeared in Journal of Algebra 236 (2001), pp.819-834.).
Abstract: We pove that the idempotent completion of a triangulated category has a (unique) structure of a triangulated category. Moreover, we show that the bounded derived category of an idempotent complete exact category is idempotent complete. Both statements were motivated by applications to motivic cohomology and algebraic K-theory. |
dvi
format.
Abstract: We define negative $K$-groups for exact categories and chain complexes in the framework of ``Frobenius pairs'' generalizing previous definitions of Bass, Karoubi, Carter, Pedersen, Thomason. We show that the classical theorems as additivity, resolution and {\it d\'evissage} (for noetherian abelian categories) continue to hold. Moreover, Thomason localization extends to negative $K$-groups, in particular, exact functors which induce equivalences of derived categories yield $K$-theory equivalences. We show that the first negative $K$-group of an abelian category vanishes and that negative $K$-groups of any noetherian abelian category vanish. Our methods yield a non-connective delooping of the $K$-theory of exact categories and chain complexes generalizing constructions of Wagoner and Pedersen-Weibel. Along the way we generalize a theorem of Auslander-Sherman proving that the K-theory homotopy fiber of E_split --> E is the K-theory of an abelian category for any exact category E and in negative degrees as well. In the appendix we replace Waldhausen's cylinder functor by a slightly weaker form of non-functorial factorization still sufficient to prove his approximation and fibration theorems. |
dvi format.
Abstract: We give a different proof of a theroem stated in a paper of Charney and Lee, "On a theorem of Giffen", which avoids a gap on page 177 of their paper. This is a preliminary version. A stylistically smoother version should appear soon on this page. |