#### Vol. 15, No. 2, 2021

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

### Appendix: Gabriele Vezzosi

Vol. 15 (2021), No. 2, 341–385
##### 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.

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