Mathematische Zeitschrift

, Volume 253, Issue 1, pp 97–134

Negative K-theory of derived categories


DOI: 10.1007/s00209-005-0889-3

Cite this article as:
Schlichting, M. Math. Z. (2006) 253: 97. doi:10.1007/s00209-005-0889-3


We define negative K-groups for exact categories and for ``derived categories'' in the framework of Frobenius pairs, generalizing definitions of Bass, Karoubi, Carter, Pedersen-Weibel and Thomason. We prove localization and vanishing theorems for these groups. Dévissage (for noetherian abelian categories), additivity, and resolution hold. We show that the first negative K-group of an abelian category vanishes, and that, in general, negative K-groups of a noetherian abelian category vanish. Our methods yield an explicit non-connective delooping of the K-theory of exact categories and chain complexes, generalizing constructions of Wagoner and Pedersen-Weibel. Extending a theorem of Auslander and Sherman, we discuss the K-theory homotopy fiber of ε→ ε and its implications for negative K-groups. In the appendix, we replace Waldhausen's cylinder functor by a slightly weaker form of non-functorial factorization which is still sufficient to prove his approximation and fibration theorems.

Mathematics Subject Classification (2000)

18E30 19D35 55P47 

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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