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The categorified Grothendieck–Riemann–Roch theorem

Part of: Higher algebraic $K$-theory (Co)homology theory

Published online by Cambridge University Press:  15 February 2021

Marc Hoyois ,
Pavel Safronov ,
Sarah Scherotzke  and
Nicolò Sibilla
Marc Hoyois
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040Regensburg, Germanymarc.hoyois@ur.de
Pavel Safronov
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8051Zurich, Switzerlandpavel.safronov@math.uzh.ch
Sarah Scherotzke
Affiliation:
Department of Mathematics, Université du Luxembourg, Maison du Nombre, 6, Avenue de la Fonte, L-4364Esch-sur-Alzette, Luxembourgsarah.scherotzke@uni.lu
Nicolò Sibilla
Affiliation:
Sissa, Via Bonomea 265, 34136Trieste TS, Italynsibilla@sissa.it; N.Sibilla@kent.ac.uk University of Kent, Canterbury, KentCT2 7NF, UK
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Abstract

In this paper we prove a categorification of the Grothendieck–Riemann–Roch theorem. Our result implies in particular a Grothendieck–Riemann–Roch theorem for Toën and Vezzosi's secondary Chern character. As a main application, we establish a comparison between the Toën–Vezzosi Chern character and the classical Chern character, and show that the categorified Chern character recovers the classical de Rham realization.

Keywords

Chern characterK-theorycategorification

MSC classification

Secondary: 14F42: Motivic cohomology; motivic homotopy theory 19D55: $K$-theory and homology; cyclic homology and cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 1 , January 2021 , pp. 154 - 214
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Copyright
© The Author(s) 2021

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References

Arinkin, D. and Gaitsgory, D., Singular support of coherent sheaves and the geometric Langlands conjecture, Selecta Math. (N.S.) 21 (2015), 1–199.CrossRefGoogle Scholar
Baas, N. A., Dundas, B. I. and Rognes, J., Two-vector bundles and forms of elliptic cohomology, in Topology, geometry and quantum field theory, ed. U. Tillmann (Cambridge University Press, Cambridge, 2004), 18–45.CrossRefGoogle Scholar
Barwick, C., On exact $\infty$-categories and the theorem of the heart, Compos. Math. 151 (2015), 2160–2186.CrossRefGoogle Scholar
Ben-Zvi, D. and Nadler, D., Loop spaces and connections, J. Topol. 5 (2012), 377–430.CrossRefGoogle Scholar
Ben-Zvi, D. and Nadler, D., Nonlinear traces, Preprint (2013), arXiv:1305.7175v3.Google Scholar
Ben-Zvi, D. and Nadler, D., Secondary traces, Preprint (2013), arXiv:1305.7177v3.Google Scholar
Bittner, F., The universal Euler characteristic for varieties of characteristic zero, Compos. Math. 140 (2004), 1011–1032.CrossRefGoogle Scholar
Blumberg, A., Gepner, D. and Tabuada, G., A universal characterization of higher algebraic K-theory, Geom. Topol. 17 (2013), 733–838.CrossRefGoogle Scholar
Bondal, A. I. and Kapranov, M. M., Representable functors, Serre functors, and mutations, Math. USSR-Izv. 35 (1990), 519.CrossRefGoogle Scholar
Bondal, A., Larsen, M. and Lunts, V., Grothendieck ring of pretriangulated categories, Int. Math. Res. Not. 29 (2004), 1461–1495.CrossRefGoogle Scholar
Borel, A., Algebraic D-modules, Perspectives in Mathematics (Academic Press, Boston, 1987).Google Scholar
Brasselet, J.-P., Schürmann, J. and Yokura, S., Hirzebruch classes and motivic Chern classes for singular spaces, J. Topol. Anal. 2 (2010), 1–55.CrossRefGoogle Scholar
Cappell, S. E. and Shaneson, J. L., Stratifiable maps and topological invariants, J. Amer. Math. Soc. 4 (1991), 521–551.CrossRefGoogle Scholar
Cisinski, D.-C. and Tabuada, G., Symmetric monoidal structures in non-commutative motives, J. K-Theory 9 (2012), 201–268.CrossRefGoogle Scholar
Drew, B., Réalisations tannakiennes des motifs mixtes triangulés, Thèse de Doctorat, Université Paris 13 (2013).Google Scholar
Gaitsgory, D., Ind-coherent sheaves, Mosc. Math. J. 13 (2013), 399–528.CrossRefGoogle Scholar
Gaitsgory, D., Sheaves of categories and the notion of 1-affineness, in Stacks and categories in geometry, topology, and algebra, Contemporary Mathematics, vol. 643 (American Mathematical Society, Providence, RI, 2015), 127–225.CrossRefGoogle Scholar
Gaitsgory, D. and Rozenblyum, N., Crystals and D-modules, Pure Appl. Math. Q. 10 (2014), 57–154.CrossRefGoogle Scholar
Gaitsgory, D. and Rozenblyum, N., A study in derived algebraic geometry, Volume I: correspondences and duality, Mathematical Surveys and Monographs, vol. 221 (American Mathematical Society, Providence, RI, 2017).CrossRefGoogle Scholar
Gaitsgory, D. and Rozenblyum, N., A study in derived algebraic geometry, Volume II: deformations, Lie theory and formal geometry, Mathematical Surveys and Monographs, vol. 221 (American Mathematical Society, Providence, RI, 2017).CrossRefGoogle Scholar
Ganter, N. and Kapranov, M., Representation and character theory in 2-categories, Adv. Math. 217 (2008), 2268–2300.CrossRefGoogle Scholar
Getzler, E., Cartan homotopy formulas and the Gauss–Manin connection in cyclic homology, in Quantum deformations of algebras and their representations, Israel Mathematics Conference Proceedings, vol. 7, eds A. Joseph and S. Snider (Bar-Ilan University, Ramat-Gan, Israel 1993), 65–78.Google Scholar
Gurski, N., Biequivalences in tricategories, Theory Appl. Categ. 26 (2012), 349–384.Google Scholar
Hall, J., Neeman, A. and Rydh, D., One positive and two negative results for derived categories of algebraic stacks, J. Inst. Math. Jussieu 18 (2019), 1087–1111.CrossRefGoogle Scholar
Hopkins, M. J., Kuhn, N. J. and Ravenel, D. C., Morava K-theories of classifying spaces and generalized characters for finite groups, in Algebraic topology homotopy and group cohomology (Springer, Berlin, 1992), 186–209.CrossRefGoogle Scholar
Hotta, R., Takeuchi, K. and Tanisaki, T., D-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236 (Birkhäuser, Boston, 2008).CrossRefGoogle Scholar
Hoyois, M., The étale symmetric Künneth theorem, Preprint (2018), arXiv:1810.00351v2.Google Scholar
Hoyois, M., Scherotzke, S. and Sibilla, N., Higher traces, noncommutative motives, and the categorified Chern character, Adv. Math. 309 (2017), 97–154.CrossRefGoogle Scholar
Johnson-Freyd, T. and Scheimbauer, C., (Op)lax natural transformations, twisted quantum field theories, and ‘even higher’ Morita categories, Adv. Math. 307 (2017), 147–223.CrossRefGoogle Scholar
Kassel, C., Cyclic homology, comodules, and mixed complexes, J. Algebra 107 (1987), 195–216.CrossRefGoogle Scholar
Keller, B., On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999), 1–56.CrossRefGoogle Scholar
Khovanov, M. and Lauda, A., A diagrammatic approach to categorification of quantum groups I, Represent. Theory Amer. Math. Soc. 13 (2009), 309–347.CrossRefGoogle Scholar
Khovanov, M. and Lauda, A., A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), 2685–2700.CrossRefGoogle Scholar
Klein, J. R., The dualizing spectrum of a topological group, Math. Ann. 319 (2001), 421–456.CrossRefGoogle Scholar
Lurie, J., Higher topos theory, Annals of Mathematics Studies, vol. 170 (Princeton University Press, Princeton, NJ, 2009).CrossRefGoogle Scholar
Lurie, J., Higher algebra (2017), http://www.math.harvard.edu/lurie/papers/HA.pdf.Google Scholar
Lurie, J., Spectral algebraic geometry (2018), http://www.math.harvard.edu/lurie/papers/SAG-rootfile.pdf.Google Scholar
MacPherson, R. D., Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432.CrossRefGoogle Scholar
Markarian, N., The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem, J. Lond. Math. Soc. (2) 79 (2009), 129–143.CrossRefGoogle Scholar
McCarthy, R., The cyclic homology of an exact category, J. Pure Appl. Algebra 93 (1994), 251–296.CrossRefGoogle Scholar
Morton, H. R., Symmetric products of the circle, Proc. Cambridge Philos. Soc. 63 (1967), 349–352.CrossRefGoogle Scholar
Orlov, D. O., Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Russ. Acad. Sci. Izv. Math. 41 (1993), 133.Google Scholar
Preygel, A., Ind-coherent complexes on loop spaces and connections, in Stacks and categories in geometry, topology, and algebra, Contemporary Mathematics, vol. 643 (American Mathematical Society, Providence, RI, 2015), 289–323.CrossRefGoogle Scholar
Pstra̧gowski, P., On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis, Master's thesis, University of Bonn (November 2014), arXiv:1411.6691.Google Scholar
Riehl, E. and Verity, D., Homotopy coherent adjunctions and the formal theory of monads, Adv. Math. 286 (2016), 802–888.CrossRefGoogle Scholar
Robalo, M., $K$-theory and the bridge from motives to noncommutative motives, Adv. Math. 269 (2015), 399–550.CrossRefGoogle Scholar
Rouquier, R., 2-Kac-Moody algebras, Preprint (2008), arXiv:0812.5023.Google Scholar
Schürmann, J., Specialization of motivic Hodge-Chern classes, Preprint (2009), arXiv:0909.3478.Google Scholar
Toën, B. and Vezzosi, G., Chern character, loop spaces and derived algebraic geometry, in Algebraic topology, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Østvær, Abel Symposia, vol. 4 (Springer, Berlin, 2009), 331–354.CrossRefGoogle Scholar
Toën, B. and Vezzosi, G., Caractères de Chern, traces équivariantes et géométrie algébrique dérivée, Selecta Math. 21 (2015), 449–554.CrossRefGoogle Scholar