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2021 Equivariant Grothendieck–Riemann–Roch and localization in operational K -theory
Dave Anderson, Richard Gonzales, Sam Payne, Gabriele Vezzosi
Algebra Number Theory 15(2): 341-385 (2021). DOI: 10.2140/ant.2021.15.341

Abstract

We produce a Grothendieck transformation from bivariant operational K-theory to Chow, with a Riemann–Roch formula that generalizes classical Grothendieck–Verdier–Riemann–Roch. We also produce Grothendieck transformations and Riemann–Roch formulas that generalize the classical Adams–Riemann–Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety X whose equivariant K-theory of vector bundles does not surject onto its ordinary K-theory, and describe the operational K-theory of spherical varieties in terms of fixed-point data.

In an appendix, Vezzosi studies operational K-theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic K-theory of relatively perfect complexes to bivariant operational K-theory.

Citation

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Dave Anderson. Richard Gonzales. Sam Payne. Gabriele Vezzosi. "Equivariant Grothendieck–Riemann–Roch and localization in operational K -theory." Algebra Number Theory 15 (2) 341 - 385, 2021. https://doi.org/10.2140/ant.2021.15.341

Information

Received: 28 June 2019; Revised: 6 May 2020; Accepted: 5 July 2020; Published: 2021
First available in Project Euclid: 23 June 2021

Digital Object Identifier: 10.2140/ant.2021.15.341

Subjects:
Primary: 19E08
Secondary: 14C35 , 14C40 , 14M25 , 14M27 , 19E20

Keywords: Bivariant theory , equivariant localization , Riemann–Roch theorems

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.15 • No. 2 • 2021
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