Vol. 15, No. 2, 2021

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Equivariant Grothendieck–Riemann–Roch and localization in operational $K$-theory

Dave Anderson, Richard Gonzales and Sam Payne

Appendix: Gabriele Vezzosi

Vol. 15 (2021), No. 2, 341–385

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.

Riemann–Roch theorems, equivariant localization, bivariant theory
Mathematical Subject Classification 2010
Primary: 19E08
Secondary: 14C35, 14C40, 14M25, 14M27, 19E20
Received: 28 June 2019
Revised: 6 May 2020
Accepted: 5 July 2020
Published: 7 April 2021
Dave Anderson
The Ohio State University
Columbus, OH
United States
Richard Gonzales
Pontificia Universidad Católica del Perú
Sam Payne
University of Texas at Austin
Austin, TX
United States
Gabriele Vezzosi
Università di Firenze