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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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