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Equivariant Grothendieck–Riemann–Roch and localization in operational K-theory
Algebra & Number Theory  (IF0.938),  Pub Date : 2021-04-07, DOI: 10.2140/ant.2021.15.341
Dave Anderson, Richard Gonzales, Sam Payne

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.