K-motives of algebraic varieties

A kind of motivic algebra of spectral categories and modules over them is developed to introduce K-motives of algebraic varieties. As an application, bivariant algebraic K-theory as well as bivariant motivic kohomology groups are defined and studied. We use Grayson's machinery to produce the Grayson motivic spectral sequence connecting bivariant K-theory to bivariant motivic kohomology. It is shown that the spectral sequence is naturally realized in the triangulated category of K-motives constructed in the paper. It is also shown that ordinary algebraic K-theory is represented by the K-motive of the point.

[1]  V. Voevodsky.  3. Cohomological Theory of Presheaves with Transfers , 2011 .

[2]  P. A. Østvær,et al.  Modules over motivic cohomology , 2008 .

[3]  Daniel Dugger.  Spectral enrichments of model categories , 2005, math/0502006.

[4]  J. Jardine.  Presheaves of Chain Complexes , 2003 .

[5]  B. Shipley,et al.  Stable model categories are categories of modules , 2003 .

[6]  A. Suslin.  On the Grayson Spectral Sequence , 2003 .

[7]  B. Dundas,et al.  Enriched functors and stable homotopy theory , 2003, Documenta Mathematica.

[8]  Philip S. Hirschhorn.  Model categories and their localizations , 2003 .

[9]  L. Hesselholt,et al.  Topological Cyclic Homology of Schemes , 2003 .

[10]  A. Suslin,et al.  The spectral sequence relating algebraic K-theory to motivic cohomology , 2002 .

[11]  Vladimir Voevodsky,et al.  Motivic cohomology groups are isomorphic to higher chow groups in any characteristic , 2002 .

[12]  Benjamin A. Blander.  Local Projective Model Structures on Simplicial Presheaves , 2001 .

[13]  A. K. Bousfield.  On the telescopic homotopy theory of spaces , 2000 .

[14]  J. Jardine.  Presheaves of symmetric spectra , 2000 .

[15]  Mark Hovey.  Spectra and symmetric spectra in general model categories , 2000, math/0004051.

[16]  Vladimir Voevodsky,et al.  Cycles, Transfers And Motivic Homology Theories , 2000 .

[17]  A. Suslin,et al.  Bloch-Kato Conjecture and Motivic Cohomology with Finite Coefficients , 2000 .

[18]  J. Jardine.  Motivic symmetric spectra , 2000, Documenta Mathematica.

[19]  A. Suslin,et al.  Relative cycles and Chow sheaves , 1999 .

[20]  B. Shipley,et al.  Algebras and Modules in Monoidal Model Categories , 1998, math/9801082.

[21]  Mark Hovey,et al.  Symmetric spectra , 1998, math/9801077.

[22]  Amnon Neeman,et al.  The Grothendieck duality theorem via Bousfield’s techniques and Brown representability , 1996 .

[23]  Daniel R. Grayson.  Weight filtrations via commuting automorphisms , 1995 .

[24]  Daniel R. Grayson.  ADAMS OPERATIONS ON HIGHER K-THEORY , 1992 .

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

[26]  Daniel R. Grayson,et al.  The loop space of the $Q$-construction , 1987 .

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

[28]  F. Waldhausen.  ALGEBRAIC K-THEORY OF SPACES I , 1978 .

[29]  Daniel R. Grayson.  Higher algebraic K-theory: II , 1976 .

[30]  B. Day.  On closed categories of functors , 1970 .

[31]  V. Voevodsky.  Cohomological theory of presheaves with transfers , 2022 .