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\(\mathbf{A}^1\)-homotopy theory of noncommutative motives. (English) Zbl 1345.14008

Summary: In this article we continue the development of a theory of noncommutative motives, initiated in [the author, Duke Math. J. 145, No. 1, 121–206 (2008; Zbl 1166.18007)]. We construct categories of \({\mathbf A}^1\)-homotopy noncommutative motives, describe their universal properties, and compute their spectra of morphisms in terms of Karoubi-Villamayor’s \(K\)-theory (\(KV\)) and Weibel’s homotopy \(K\)-theory (\(KH\)). As an application, we obtain a complete classification of all the natural transformations defined on \(KV,\;KH\). This leads to a streamlined construction of Weibel’s homotopy Chern character from \(KV\) to periodic cyclic homology. Along the way we extend Dwyer-Friedlander’s étale \(K\)-theory to the noncommutative world, and develop the universal procedure of forcing a functor to preserve filtered homotopy colimits.

MSC:

14A22 Noncommutative algebraic geometry
14C15 (Equivariant) Chow groups and rings; motives
16E20 Grothendieck groups, \(K\)-theory, etc.
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
19D35 Negative \(K\)-theory, NK and Nil
19D55 \(K\)-theory and homology; cyclic homology and cohomology
19L10 Riemann-Roch theorems, Chern characters

Citations:

Zbl 1166.18007
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References:

[1] J. Adamek and J. Rosicky, Locally presentable and accessible categories. London Mathematical Society Lecture Note Series, 189. Cambridge University Press, Cambridge, 1994. · Zbl 0795.18007
[2] A. Blumberg and M. Mandell, Localization theorems in topological Hochschild homology and topological cyclic homology. Geom. Topol. 16 (2012), no. 2, 1053-1120. · Zbl 1282.19004 · doi:10.2140/gt.2012.16.1053
[3] A. K. Bousfield and E. M. Friedlander, Homotopy theory of EUR-spaces, spectra, and bisimplicial sets. Geometric applications of homotopy theory (Proc. Conf., Evanston, IlI, 1977), II, 80-130, Lecture Notes in Math. 658, Springer, Berlin, 1978. · Zbl 0405.55021
[4] D.-C. Cisinski and G. Tabuada, Symmetric monoidal structure on noncommu- tative motives. Journal of K-Theory 9 (2012), no. 2, 201-268. · Zbl 1256.19002 · doi:10.1017/is011011005jkt169
[5] D.-C. Cisinski and G. Tabuada, Nonconnective K-theory via universal invariants. Compositio Mathematica 147 (2011), 1281-1320. · Zbl 1247.19001 · doi:10.1112/S0010437X11005380
[6] V. Drinfeld, DG quotients of DG categories. J. of Algebra 272 (2004), 643-691. · Zbl 1064.18009 · doi:10.1016/j.jalgebra.2003.05.001
[7] W. Dwyer and E. Friedlander, Algebraic and étale K-theory. Trans. of AMS 292 (1985), no. 1, 247-280. · Zbl 0581.14012 · doi:10.2307/2000179
[8] W. Dwyer and E. Friedlander, Étale K-theory and arithmetic. Bulletin AMS (N.S.) 6 (1982), no. 3, 453-455. · Zbl 0494.18009 · doi:10.1090/S0273-0979-1982-15013-3
[9] E. Friedlander, Étale K-theory. I. Connections with étale cohomology and algebraic vector bundles. Invent. Math. 60 (1980), no. 2, 105-134. · Zbl 0519.14010 · doi:10.1007/BF01405150
[10] E. Friedlander, Étale K-theory. II. Connections with algebraic K-theory. Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 231-256. · Zbl 0537.14011
[11] T. Goodwillie, Cyclic homology, derivations, and the free loopspace. Topology 24 (1985), no. 2, 187-215. · Zbl 0569.16021 · doi:10.1016/0040-9383(85)90055-2
[12] P. Goerss and J. Jardine, Simplicial homotopy theory. Progress in Mathematics 174. Birkhäuser Verlag, Basel, 1999. · Zbl 0949.55001
[13] A. Grothendieck, Les Dérivateurs. Available at:
[14] P. Hirschhorn, Model categories and their localizations. Math. Surveys and Monographs 99. American Mathematical Society, Providence, RI, 2003. · Zbl 1017.55001
[15] M. Hovey, Model categories. Math. Surveys and Monographs, 63. American Mathematical Society, Providence, RI, 1999. · Zbl 0909.55001
[16] M. Hovey, B. Shipley and J. Smith, Symmetric spectra. J. AMS 13 (2000), no. 1, 149-208. · Zbl 0931.55006 · doi:10.1090/S0894-0347-99-00320-3
[17] C. Kassel, Cyclic homology, Comodules, and Mixed complexes. J. of Alg. 107 (1987), 195-216. · Zbl 0617.16015 · doi:10.1016/0021-8693(87)90086-X
[18] M. Karoubi and O. Villamayor, K-théorie algébrique et K-théorie topologique I . Math. Scand. 28 (1971), 265-307. · Zbl 0231.18018
[19] M. Karoubi and O. Villamayor, K-théorie algébrique et K-théorie topologique II . Math. Scand. 32 (1973), 57-86. · Zbl 0272.18009
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