We produce a Grothendieck transformation from bivariant operational -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 whose equivariant -theory of vector bundles does not surject onto its ordinary -theory, and describe the operational -theory of spherical varieties in terms of fixed-point data.
In an appendix, Vezzosi studies operational -theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic -theory of relatively perfect complexes to bivariant operational -theory.
"Equivariant Grothendieck–Riemann–Roch and localization in operational -theory." Algebra Number Theory 15 (2) 341 - 385, 2021. https://doi.org/10.2140/ant.2021.15.341