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

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