The fundamental theorem via derived Morita invariance, localization, and A(1)-homotopy invariance

We prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and A^1-homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel's fundamental theorems in homotopy algebraic K-theory, and periodic cyclic homology, respectively.

[1]  Denis-Charles Cisinski,et al.  Symmetric monoidal structure on non-commutative motives , 2010, Journal of K-Theory.

[2]  Denis-Charles Cisinski,et al.  Non-connective K-theory via universal invariants , 2009, Compositio Mathematica.

[3]  D. Kaledin,et al.  Motivic structures in non-commutative geometry , 2010, 1003.3210.

[4]  D. Orlov,et al.  Uniqueness of enhancement for triangulated categories , 2009, 0908.4187.

[5]  M. Kontsevich.  Notes on Motives in Finite Characteristic , 2007, math/0702206.

[6]  Gonçalo Tabuada.  Higher K-theory via universal invariants , 2007, 0706.2420.

[7]  M. Schlichting.  Negative K-theory of derived categories , 2006 .

[8]  Bernhard Keller,et al.  On differential graded categories , 2006, math/0601185.

[9]  Gonçalo Tabuada.  Invariants additifs de dg-catégories , 2005, math/0507227.

[10]  V. Drinfeld.  DG quotients of DG categories , 2002, math/0210114.

[11]  M. Bergh,et al.  Generators and representability of functors in commutative and noncommutative geometry , 2002, math/0204218.

[12]  C. Weibel,et al.  An Introduction to Homological Algebra: References , 1960 .

[13]  Bernhard Keller,et al.  Invariance and localization for cyclic homology of DG algebras , 1998 .

[14]  R. Thomason,et al.  Higher Algebraic K-Theory of Schemes and of Derived Categories , 1990 .

[15]  Christian Kassel.  Cyclic homology, comodules, and mixed complexes , 1987 .

[16]  B. Feigin,et al.  Additive K-theory and crystalline cohomology , 1985 .

[17]  C. Weibel.  KV-theory of categories , 1981 .

[18]  A. K. Bousfield,et al.  Homotopy theory of Γ-spaces, spectra, and bisimplicial sets , 1978 .

[19]  Séminaire Bourbaki,et al.  Dix exposés sur la cohomologie des schémas , 1968 .