Systems of Diagram Categories and K-theory Ii

The additivity theorem for comlicial dérivateurs is proved. Also, to any exact category in the sense of Quillen a K-theory space is associated. This K-theory is shown to satisfy the additivity, approximation and resolution theorems. CONTENTS 1. Introduzione 1 2. Chain complexes, homotopies, derived categories 3 2.1. Definition of the derived category 3 2.2. Homotopy pullbacks and homotopy pushouts 4 2.3. Getting rid of homotopy commutative squares 5 3. The additivity theorem 9 4. Complicial dérivateurs 20 5. The derived K-theory of an exact category 24 5.1. Approximation and Resolution theorems 25 5.2. Pairings 31 5.3. Conjectures 32 6. Addendum 33 6.1. The axioms 34 6.2. The S.-construction and K-theory space 37 References 38 1. INTRODUZIONE It is well known due most recently to work of Schlichting [18] that, in general, there is noK-theory for triangulated categories satisfying localization and reconstructing Quillen’s K-theory of an exact category from its derived category. There are two approaches to replace the naive notion of derived category by something richer, from which the K-theory might be obtained by some explicit construction. One approach, suggested by Dwyer Date: January 26th, 2004. 2000 Mathematics Subject Classification. Primary 19D99.