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Topological K-theory of complex noncommutative spaces

Part of: Algebraic geometry: Foundations Higher algebraic $K$-theory

Published online by Cambridge University Press:  22 September 2015

Anthony Blanc
Anthony Blanc*
Affiliation:
Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany email anthony.blanc@ihes.fr
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Abstract

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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.

Keywords

noncommutative algebraic geometrydg-categoriesK-theorycyclic homology

MSC classification

Primary: 14A22: Noncommutative algebraic geometry 19D55: $K$-theory and homology; cyclic homology and cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 3 , March 2016 , pp. 489 - 555
Copyright
© The Author 2015 

References

Baker, A. and Richter, B., Uniqueness of E infinity structures for connective covers, Proc. Amer. Math. Soc. 136 (2008), 707–714.CrossRefGoogle Scholar
Barwick, C., On (enriched) left Bousfield localization of model categories, Preprint (2007),arXiv:0708.2067.Google Scholar
Bass, H., Algebraic K-theory (WA Benjamin, New York, 1968).Google Scholar
Blanc, A., Invariants topologiques des espaces non commutatifs, Preprint (2013),arXiv:1307.6430, PhD thesis, Université Montpellier 2 (in French).Google Scholar
Blander, B. A., Local projective model structures on simplicial presheaves, J. K-Theory 24 (2001), 283–301.Google Scholar
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.Google Scholar
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), 80–130.Google Scholar
Cisinski, D.-C., Descente par éclatements en K-théorie invariante par homotopie, Ann. of Math. (2) 177 (2013), 425–448.CrossRefGoogle Scholar
Cisinski, D.-C. and Tabuada, G., Non-connective K-theory via universal invariants, Compositio Math. 147 (2011), 1281–1320.Google Scholar
Cisinski, D.-C. and Tabuada, G., Symmetric monoidal structure on non-commutative motives, J. K-Theory 9 (2012), 201–268.Google Scholar
Cortiñas, G., Haesemeyer, C., Schlichting, M. and Weibel, C., Cyclic homology, cdh-cohomology and negative K-theory, Ann. of Math. (2) (2008), 549–573.CrossRefGoogle Scholar
Deligne, P., Théorie de Hodge III, Publ. Math. Inst. Hautes Études Sci. 44 (1974), 5–77.Google Scholar
Dugger, D., Universal homotopy theories, Adv. Math. 164 (2001), 144–176.Google Scholar
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), 9–51.Google Scholar
Dugger, D. and Isaksen, D. C., Hypercovers in topology, Preprint (2001), arXiv:math/0111287.Google Scholar
Dyckerhoff, T., Compact generators in categories of matrix factorizations, Duke Math. J. 159 (2011), 223–274.CrossRefGoogle Scholar
Efimov, A. I., Cyclic homology of categories of matrix factorizations, Preprint (2012),arXiv:1212.2859.Google Scholar
Esnault, H. and Viehweg, E., Deligne–Beilinson cohomology (Max-Planck-Institut für Mathematik, 1987).Google Scholar
Freed, D. S., Remarks on Chern–Simons theory, Bull. Amer. Math. Soc. (N.S.) 46 (2009), 221–254.Google Scholar
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).Google Scholar
Friedlander, E. and Walker, M., Comparing K-theories for complex varieties, Amer. J. Math. (2001), 779–810.CrossRefGoogle Scholar
Friedlander, E. and Walker, M., Rational isomorphisms between K-theories and cohomology theories, Invent. Math. 154 (2003), 1–61.Google Scholar
Friedlander, E. and Walker, M., Semi-topological K-theory, in Handbook of K-theory, eds Friedlander, E. and Grayson, D. R. (Springer, Berlin, 2005), 877–924.Google Scholar
Fukaya, K., Oh, Y.-G., Ohta, H. and Ono, K., Lagrangian intersection Floer theory (American Mathematical Society, Providence, RI, 2000).Google Scholar
Goodwillie, T. G., Cyclic homology, derivations, and the free loopspace, Topology 24 (1985), 187–215.Google Scholar
Haesemeyer, C., Descent properties of homotopy K-theory, Duke Math. J. 125 (2004), 589–619.Google Scholar
Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero: II, Ann. of Math. (2) 79 (1964), 205–326.Google Scholar
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), 165–185.Google Scholar
Hirschhorn, P. S., Model categories and their localizations (American Mathematical Society, Providence, RI, 2009).Google Scholar
Hochschild, G., Kostant, B. and Rosenberg, A., Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962), 383–408.Google Scholar
Hovey, M., Monoidal model categories, Preprint (1998), arXiv:math/9803002.Google Scholar
Hovey, M., Model categories, Mathematical Surveys and Monographs, vol. 63 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Hovey, M., Model category structures on chain complexes of sheaves, Trans. Amer. Math. Soc. 353 (2001), 2441–2457.CrossRefGoogle Scholar
Hovey, M., Shipley, B. and Smith, J., Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149–208.Google Scholar
Jardine, J. F., Simplical presheaves, J. Pure Appl. Algebra 47 (1987), 35–87.Google Scholar
Kaledin, D., Motivic structures in non-commutative geometry, in Proceedings of the International Congress of Mathematicians 2010 (Hindustan Book Agency, New Delhi, 2011), 461–496.Google Scholar
Kassel, C., Cyclic homology, comodules, and mixed complexes, J. Algebra 107 (1987), 195–216.Google Scholar
Katzarkov, L., Kontsevich, M. and Pantev, T., Hodge theoretic aspects of mirror symmetry, Preprint (2008), arXiv:0806.0107.Google Scholar
Katzarkov, L., Pantev, T. and Toën, B., Algebraic and topological aspects of the schematization functor, Compositio Math. 145 (2009), 633–686.Google Scholar
Keller, B., Invariance and localization for cyclic homology of dg algebras, J. Pure Appl. Algebra 123 (1998), 223–273.Google Scholar
Keller, B., On the cyclic homology of ringed spaces and schemes, Doc. Math 3 (1998), 231–259.Google Scholar
Keller, B., On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999), 1–56.Google Scholar
Keller, B., Cluster algebras, quiver representations and triangulated categories, Preprint (2008), arXiv:0807.1960.Google Scholar
Kontsevich, M., Deformation quantization of algebraic varieties, Lett. Math. Phys. 56 (2001), 271–294.Google Scholar
Krause, H., Representations of quivers via reflection functors, Preprint (2008),arXiv:0804.1428.Google Scholar
Loday, J.-L., Cyclic homology, Grundlehren der mathematischen Wissenschaften, vol. 301 (Springer, Berlin, 1998).Google Scholar
Lurie, J., Higher topos theory (Princeton University Press, Princeton, NJ, 2009).Google Scholar
Marcolli, M. and Tabuada, G., Jacobians of noncommutative motives, Preprint (2012),arXiv:1212.1118.Google Scholar
Morel, F. and Voevodsky, V., A1-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45–143.Google Scholar
Orlov, D. O., Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys 58 (2003), 511–591.Google Scholar
Orlov, D. O., Derived categories of coherent sheaves and triangulated categories of singularities, Preprint (2005), arXiv:math/0503632.Google Scholar
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), 85–147.Google Scholar
Riou, J., Dualité de Spanier–Whitehead en géométrie algébrique, C. R. Math. 340 (2005), 431–436.Google Scholar
Robalo, M., Noncommutative motives I: A universal characterization of the motivic stable homotopy theory of schemes, Preprint (2012), arXiv:1206.3645.Google Scholar
Robalo, M., Noncommutative motives II: K-theory and noncommutative motives, Preprint (2013), arXiv:1306.3795.Google Scholar
Rosenberg, J., Algebraic K-theory and its applications, Graduate Texts in Mathematics, vol. 147 (Springer, New York, 1994).Google Scholar
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), 259–271; MR 2397629 (2009f:53150).Google Scholar
Schlichting, M., Negative K-theory of derived categories, Math. Z. 253 (2006), 97–134.Google Scholar
Schwede, S., An untitled book project about symmetric spectra, available at: http://www.math.uni-bonn.de/people/schwede/SymSpec.pdf.Google Scholar
Schwede, S. and Shipley, B. E., Algebras and modules in monoidal model categories, Proc. Lond. Math. Soc. (3) 80 (2000), 491–511.Google Scholar
Schwede, S. and Shipley, B. E., Stable model categories are categories of modules, Topology 42 (2003), 103–153.Google Scholar
Segal, G., Categories and cohomology theories, Topology 13 (1974), 293–312.Google Scholar
Seidel, P., Fukaya categories and Picard–Lefschetz theory, Advances in Mathematics (European Mathematical Society, Zürich, 2008).CrossRefGoogle Scholar
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.Google Scholar
Simpson, C., The topological realization of a simplicial presheaf, Preprint (1996),arXiv:q-alg/9609004.Google Scholar
Suslin, A. and Voevodsky, V., Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), 61–94.Google Scholar
Tabuada, G., Théorie homotopique des dg-catégories, PhD thesis, Preprint (2007),arXiv:0710.4303v1 [math.KT].Google Scholar
Tabuada, G., Higher K-theory via universal invariants, Duke Math. J. 145 (2008), 121–206.Google Scholar
Tabuada, G., Products, multiplicative Chern characters, and finite coefficients via noncommutative motives, J. Pure Appl. Algebra 217 (2013), 1279–1293.Google Scholar
Thomason, R. W., Algebraic K-theory and étale cohomology, Ann. Sci. Éc. Norm. Supér. (4) 18 (1985), 437–552.Google Scholar
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), 247–435.Google Scholar
Toën, B., Vers une interprétation galoisienne de la théorie de l’homotopie, Cah. Topol. Géom. Différ. Catég. 43 (2002), 257–312.Google Scholar
Toën, B., Derived Hall algebras, Duke Math. J. 135 (2006), 587–615.Google Scholar
Toën, B., The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615–667.Google Scholar
B. Toën, Lectures on saturated $dg$-categories (handwritten notes by D. Auroux)http://www-math.mit.edu/∼auroux/frg/miami10-notes, January 2010.Google Scholar
Toën, B. and Vaquié, M., Moduli of objects in dg-categories, Ann. Sci. Éc. Norm. Supér. (4) 40 (2007), 387–444.Google Scholar
Toën, B. and Vezzosi, G., Homotopical algebraic geometry I: Topos theory, Adv. Math. 193(902) (2005), 257–372.Google Scholar
Toën, B. and Vezzosi, G., Homotopical algebraic geometry II: Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008).Google Scholar
Toën, B. and Vezzosi, G., Caractères de Chern, traces équivariantes et géométrie algébrique dérivée, Preprint (2009), arXiv:0903.3292.Google Scholar
Tsygan, B., On the Gauss–Manin connection in cyclic homology, Methods Funct. Anal. Topology 13 (2007), 83–94.Google Scholar
Weibel, C., Cyclic homology for schemes, Proc. Amer. Math. Soc. 124 (1996), 1655–1662.Google Scholar
Weibel, C., The Hodge filtration and cyclic homology, J. K-Theory 12 (1997), 145–164.Google Scholar