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Topological K-theory of complex noncommutative spaces

  • Anthony Blanc (a1)

The purpose of this work is to give a definition of a topological K-theory for dg-categories over $\mathbb{C}$ and to prove that the Chern character map from algebraic K-theory to periodic cyclic homology descends naturally to this new invariant. This topological Chern map provides a natural candidate for the existence of a rational structure on the periodic cyclic homology of a smooth proper dg-algebra, within the theory of noncommutative Hodge structures. The definition of topological K-theory consists in two steps: taking the topological realization of algebraic K-theory and inverting the Bott element. The topological realization is the left Kan extension of the functor ‘space of complex points’ to all simplicial presheaves over complex algebraic varieties. Our first main result states that the topological K-theory of the unit dg-category is the spectrum $\mathbf{BU}$ . For this we are led to prove a homotopical generalization of Deligne’s cohomological proper descent, using Lurie’s proper descent. The fact that the Chern character descends to topological K-theory is established by using Kassel’s Künneth formula for periodic cyclic homology and the proper descent. In the case of a dg-category of perfect complexes on a separated scheme of finite type, we show that we recover the usual topological K-theory of complex points. We show as well that the Chern map tensorized with $\mathbb{C}$ is an equivalence in the case of a finite-dimensional associative algebra – providing a formula for the periodic homology groups in terms of the stack of finite-dimensional modules.

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[BR08]Baker, A. and Richter, B., Uniqueness of E infinity structures for connective covers, Proc. Amer. Math. Soc. 136 (2008), 707714.
[Bar07]Barwick, C., On (enriched) left Bousfield localization of model categories, Preprint (2007),arXiv:0708.2067.
[Bas68]Bass, H., Algebraic K-theory (WA Benjamin, New York, 1968).
[Bla13]Blanc, A., Invariants topologiques des espaces non commutatifs, Preprint (2013),arXiv:1307.6430, PhD thesis, Université Montpellier 2 (in French).
[Bla01]Blander, B. A., Local projective model structures on simplicial presheaves, J. K-Theory 24 (2001), 283301.
[BO02]Bondal, A. I. and Orlov, D. O., Derived categories of coherent sheaves, in Proceedings of the International Congress of Mathematicians 2002 (Higher Education Press, Beijing, 2002), 47.
[BF78]Bousfield, A. and Friedlander, E., Homotopy theory of  Γ-spaces, spectra, and bisimplicial sets, in Geometric applications of homotopy theory, II, Lecture Notes in Mathematics, 658 (Springer, Berlin, 1978), 80130.
[Cis13]Cisinski, D.-C., Descente par éclatements en K-théorie invariante par homotopie, Ann. of Math. (2) 177 (2013), 425448.
[CT11]Cisinski, D.-C. and Tabuada, G., Non-connective K-theory via universal invariants, Compositio Math. 147 (2011), 12811320.
[CT12]Cisinski, D.-C. and Tabuada, G., Symmetric monoidal structure on non-commutative motives, J. K-Theory 9 (2012), 201268.
[CHSW08]Cortiñas, G., Haesemeyer, C., Schlichting, M. and Weibel, C., Cyclic homology, cdh-cohomology and negative K-theory, Ann. of Math. (2) (2008), 549573.
[Del74]Deligne, P., Théorie de Hodge III, Publ. Math. Inst. Hautes Études Sci. 44 (1974), 577.
[Dug01]Dugger, D., Universal homotopy theories, Adv. Math. 164 (2001), 144176.
[DHI04]Dugger, D., Hollander, S. and Isaksen, D. C., Hypercovers and simplicial presheaves, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136 (Cambridge University Press, 2004), 951.
[DI01]Dugger, D. and Isaksen, D. C., Hypercovers in topology, Preprint (2001), arXiv:math/0111287.
[Dyc11]Dyckerhoff, T., Compact generators in categories of matrix factorizations, Duke Math. J. 159 (2011), 223274.
[Efi12]Efimov, A. I., Cyclic homology of categories of matrix factorizations, Preprint (2012),arXiv:1212.2859.
[EV87]Esnault, H. and Viehweg, E., Deligne–Beilinson cohomology (Max-Planck-Institut für Mathematik, 1987).
[Fre09]Freed, D. S., Remarks on Chern–Simons theory, Bull. Amer. Math. Soc. (N.S.) 46 (2009), 221254.
[FM94]Friedlander, E. M. and Mazur, B., Filtrations on the homology of algebraic varieties, Memoirs of the American Mathematical Society, vol. 529 (American Mathematical Society, Providence, RI, 1994).
[FW01]Friedlander, E. and Walker, M., Comparing K-theories for complex varieties, Amer. J. Math. (2001), 779810.
[FW03]Friedlander, E. and Walker, M., Rational isomorphisms between K-theories and cohomology theories, Invent. Math. 154 (2003), 161.
[FW05]Friedlander, E. and Walker, M., Semi-topological K-theory, in Handbook of K-theory, eds Friedlander, E. and Grayson, D. R. (Springer, Berlin, 2005), 877924.
[FOOO00]Fukaya, K., Oh, Y.-G., Ohta, H. and Ono, K., Lagrangian intersection Floer theory (American Mathematical Society, Providence, RI, 2000).
[Goo85]Goodwillie, T. G., Cyclic homology, derivations, and the free loopspace, Topology 24 (1985), 187215.
[Hae04]Haesemeyer, C., Descent properties of homotopy K-theory, Duke Math. J. 125 (2004), 589619.
[Hir64]Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero: II, Ann. of Math. (2) 79 (1964), 205326.
[Hir75]Hironaka, H., Triangulations of algebraic sets, in Algebraic geometry (Proceedings of Symposia in Pure Mathematics, Vol. 29, Humboldt State University, Arcata, California, 1974) (American Mathematical Society, Providence, RI, 1975), 165185.
[Hir09]Hirschhorn, P. S., Model categories and their localizations (American Mathematical Society, Providence, RI, 2009).
[HKR62]Hochschild, G., Kostant, B. and Rosenberg, A., Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962), 383408.
[Hov98]Hovey, M., Monoidal model categories, Preprint (1998), arXiv:math/9803002.
[Hov99]Hovey, M., Model categories, Mathematical Surveys and Monographs, vol. 63 (American Mathematical Society, Providence, RI, 1999).
[Hov01]Hovey, M., Model category structures on chain complexes of sheaves, Trans. Amer. Math. Soc. 353 (2001), 24412457.
[HSS00]Hovey, M., Shipley, B. and Smith, J., Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149208.
[Jar87]Jardine, J. F., Simplical presheaves, J. Pure Appl. Algebra 47 (1987), 3587.
[Kal10]Kaledin, D., Motivic structures in non-commutative geometry, in Proceedings of the International Congress of Mathematicians 2010 (Hindustan Book Agency, New Delhi, 2011), 461496.
[Kas87]Kassel, C., Cyclic homology, comodules, and mixed complexes, J. Algebra 107 (1987), 195216.
[KKP08]Katzarkov, L., Kontsevich, M. and Pantev, T., Hodge theoretic aspects of mirror symmetry, Preprint (2008), arXiv:0806.0107.
[KPT09]Katzarkov, L., Pantev, T. and Toën, B., Algebraic and topological aspects of the schematization functor, Compositio Math. 145 (2009), 633686.
[Kel98a]Keller, B., Invariance and localization for cyclic homology of dg algebras, J. Pure Appl. Algebra 123 (1998), 223273.
[Kel98b]Keller, B., On the cyclic homology of ringed spaces and schemes, Doc. Math 3 (1998), 231259.
[Kel99]Keller, B., On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999), 156.
[Kel08]Keller, B., Cluster algebras, quiver representations and triangulated categories, Preprint (2008), arXiv:0807.1960.
[Kon01]Kontsevich, M., Deformation quantization of algebraic varieties, Lett. Math. Phys. 56 (2001), 271294.
[Kra08]Krause, H., Representations of quivers via reflection functors, Preprint (2008),arXiv:0804.1428.
[Lod98]Loday, J.-L., Cyclic homology, Grundlehren der mathematischen Wissenschaften, vol. 301 (Springer, Berlin, 1998).
[Lur09]Lurie, J., Higher topos theory (Princeton University Press, Princeton, NJ, 2009).
[MT12]Marcolli, M. and Tabuada, G., Jacobians of noncommutative motives, Preprint (2012),arXiv:1212.1118.
[MV99]Morel, F. and Voevodsky, V., A1-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45143.
[Orl03]Orlov, D. O., Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys 58 (2003), 511591.
[Orl05]Orlov, D. O., Derived categories of coherent sheaves and triangulated categories of singularities, Preprint (2005), arXiv:math/0503632.
[Qui]Quillen, D., Higher algebraic K-theory. I, in Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Mathematics, vol. 341 (Springer, Berlin, 1973), 85147.
[Rio05]Riou, J., Dualité de Spanier–Whitehead en géométrie algébrique, C. R. Math. 340 (2005), 431436.
[Rob12]Robalo, M., Noncommutative motives I: A universal characterization of the motivic stable homotopy theory of schemes, Preprint (2012), arXiv:1206.3645.
[Rob13]Robalo, M., Noncommutative motives II: K-theory and noncommutative motives, Preprint (2013), arXiv:1306.3795.
[Ros94]Rosenberg, J., Algebraic K-theory and its applications, Graduate Texts in Mathematics, vol. 147 (Springer, New York, 1994).
[Sch08]Schapira, P., Deformation quantization modules on complex symplectic manifolds, in Poisson geometry in mathematics and physics, Contemporary Mathematics, vol. 450 (American Mathematical Society, Providence, RI, 2008), 259271; MR 2397629 (2009f:53150).
[Sch06]Schlichting, M., Negative K-theory of derived categories, Math. Z. 253 (2006), 97134.
[Sch]Schwede, S., An untitled book project about symmetric spectra, available at:
[SS00]Schwede, S. and Shipley, B. E., Algebras and modules in monoidal model categories, Proc. Lond. Math. Soc. (3) 80 (2000), 491511.
[SS03]Schwede, S. and Shipley, B. E., Stable model categories are categories of modules, Topology 42 (2003), 103153.
[Seg74]Segal, G., Categories and cohomology theories, Topology 13 (1974), 293312.
[Sei08]Seidel, P., Fukaya categories and Picard–Lefschetz theory, Advances in Mathematics (European Mathematical Society, Zürich, 2008).
[SGA4]Artin, M., Grothendieck, A. and Verdier, J. L., Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, vol. 269 (Springer, Berlin, 1972), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier, avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat.
[Sim96]Simpson, C., The topological realization of a simplicial presheaf, Preprint (1996),arXiv:q-alg/9609004.
[SV96]Suslin, A. and Voevodsky, V., Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), 6194.
[Tab07]Tabuada, G., Théorie homotopique des dg-catégories, PhD thesis, Preprint (2007),arXiv:0710.4303v1 [math.KT].
[Tab08]Tabuada, G., Higher K-theory via universal invariants, Duke Math. J. 145 (2008), 121206.
[Tab13]Tabuada, G., Products, multiplicative Chern characters, and finite coefficients via noncommutative motives, J. Pure Appl. Algebra 217 (2013), 12791293.
[Tho85]Thomason, R. W., Algebraic K-theory and étale cohomology, Ann. Sci. Éc. Norm. Supér. (4) 18 (1985), 437552.
[TT90]Thomason, R. W. and Trobaugh, T., Higher algebraic K-theory of schemes and of derived categories, in The Grothendieck Festschrift, Vol. III, Progress in Mathematics, vol. 88 (Birkhäuser, Boston, 1990), 247435.
[Toë02]Toën, B., Vers une interprétation galoisienne de la théorie de l’homotopie, Cah. Topol. Géom. Différ. Catég. 43 (2002), 257312.
[Toë06]Toën, B., Derived Hall algebras, Duke Math. J. 135 (2006), 587615.
[Toë07]Toën, B., The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615667.
[Toë10]B. Toën, Lectures on saturated -categories (handwritten notes by D. Auroux)∼auroux/frg/miami10-notes, January 2010.
[TVa07]Toën, B. and Vaquié, M., Moduli of objects in dg-categories, Ann. Sci. Éc. Norm. Supér. (4) 40 (2007), 387444.
[TVe05]Toën, B. and Vezzosi, G., Homotopical algebraic geometry I: Topos theory, Adv. Math. 193(902) (2005), 257372.
[TVe08]Toën, B. and Vezzosi, G., Homotopical algebraic geometry II: Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008).
[TVe09]Toën, B. and Vezzosi, G., Caractères de Chern, traces équivariantes et géométrie algébrique dérivée, Preprint (2009), arXiv:0903.3292.
[Tsy07]Tsygan, B., On the Gauss–Manin connection in cyclic homology, Methods Funct. Anal. Topology 13 (2007), 8394.
[Wei96]Weibel, C., Cyclic homology for schemes, Proc. Amer. Math. Soc. 124 (1996), 16551662.
[Wei97]Weibel, C., The Hodge filtration and cyclic homology, J. K-Theory 12 (1997), 145164.
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