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Topological K-theory of complex noncommutative spaces

  • Anthony Blanc (a1)
Abstract

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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A. Baker and B. Richter , Uniqueness of E infinity structures for connective covers, Proc. Amer. Math. Soc. 136 (2008), 707714.

A. Bousfield and E. Friedlander , 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.

D.-C. Cisinski , Descente par éclatements en K-théorie invariante par homotopie, Ann. of Math. (2) 177 (2013), 425448.

G. Cortiñas , C. Haesemeyer , M. Schlichting and C. Weibel , Cyclic homology, cdh-cohomology and negative K-theory, Ann. of Math. (2) (2008), 549573.

P. Deligne , Théorie de Hodge III, Publ. Math. Inst. Hautes Études Sci. 44 (1974), 577.

D. Dugger , Universal homotopy theories, Adv. Math. 164 (2001), 144176.

T. Dyckerhoff , Compact generators in categories of matrix factorizations, Duke Math. J. 159 (2011), 223274.

D. S. Freed , Remarks on Chern–Simons theory, Bull. Amer. Math. Soc. (N.S.) 46 (2009), 221254.

E. Friedlander and M. Walker , Comparing K-theories for complex varieties, Amer. J. Math. (2001), 779810.

E. Friedlander and M. Walker , Rational isomorphisms between K-theories and cohomology theories, Invent. Math. 154 (2003), 161.

E. Friedlander and M. Walker , Semi-topological K-theory, in Handbook of K-theory, eds E. Friedlander and D. R. Grayson (Springer, Berlin, 2005), 877924.

T. G. Goodwillie , Cyclic homology, derivations, and the free loopspace, Topology 24 (1985), 187215.

C. Haesemeyer , Descent properties of homotopy K-theory, Duke Math. J. 125 (2004), 589619.

H. Hironaka , Resolution of singularities of an algebraic variety over a field of characteristic zero: II, Ann. of Math. (2) 79 (1964), 205326.

P. S. Hirschhorn , Model categories and their localizations (American Mathematical Society, Providence, RI, 2009).

G. Hochschild , B. Kostant and A. Rosenberg , Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962), 383408.

M. Hovey , Model category structures on chain complexes of sheaves, Trans. Amer. Math. Soc. 353 (2001), 24412457.

M. Hovey , B. Shipley and J. Smith , Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149208.

J. F. Jardine , Simplical presheaves, J. Pure Appl. Algebra 47 (1987), 3587.

D. Kaledin , Motivic structures in non-commutative geometry, in Proceedings of the International Congress of Mathematicians 2010 (Hindustan Book Agency, New Delhi, 2011), 461496.

C. Kassel , Cyclic homology, comodules, and mixed complexes, J. Algebra 107 (1987), 195216.

B. Keller , Invariance and localization for cyclic homology of dg algebras, J. Pure Appl. Algebra 123 (1998), 223273.

B. Keller , On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999), 156.

M. Kontsevich , Deformation quantization of algebraic varieties, Lett. Math. Phys. 56 (2001), 271294.

J.-L. Loday , Cyclic homology, Grundlehren der mathematischen Wissenschaften, vol. 301 (Springer, Berlin, 1998).

F. Morel and V. Voevodsky , A1-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45143.

D. O. Orlov , Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys 58 (2003), 511591.

J. Riou , Dualité de Spanier–Whitehead en géométrie algébrique, C. R. Math. 340 (2005), 431436.

J. Rosenberg , Algebraic K-theory and its applications, Graduate Texts in Mathematics, vol. 147 (Springer, New York, 1994).

P. Schapira , 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).

M. Schlichting , Negative K-theory of derived categories, Math. Z. 253 (2006), 97134.

S. Schwede and B. E. Shipley , Algebras and modules in monoidal model categories, Proc. Lond. Math. Soc. (3) 80 (2000), 491511.

S. Schwede and B. E. Shipley , Stable model categories are categories of modules, Topology 42 (2003), 103153.

G. Segal , Categories and cohomology theories, Topology 13 (1974), 293312.

P. Seidel , Fukaya categories and Picard–Lefschetz theory, Advances in Mathematics (European Mathematical Society, Zürich, 2008).

A. Suslin and V. Voevodsky , Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), 6194.

G. Tabuada , Higher K-theory via universal invariants, Duke Math. J. 145 (2008), 121206.

G. Tabuada , Products, multiplicative Chern characters, and finite coefficients via noncommutative motives, J. Pure Appl. Algebra 217 (2013), 12791293.

R. W. Thomason and T. Trobaugh , 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.

B. Toën , Derived Hall algebras, Duke Math. J. 135 (2006), 587615.

B. Toën , The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615667.

B. Toën and G. Vezzosi , Homotopical algebraic geometry I: Topos theory, Adv. Math. 193(902) (2005), 257372.

C. Weibel , Cyclic homology for schemes, Proc. Amer. Math. Soc. 124 (1996), 16551662.

C. Weibel , The Hodge filtration and cyclic homology, J. K-Theory 12 (1997), 145164.

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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