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

Dave Anderson, Richard Gonzales, Sam Payne, Gabriele Vezzosi

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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.

Original languageEnglish
Pages (from-to)341-385
Number of pages45
JournalAlgebra and Number Theory
Volume15
Issue number2
DOIs
StatePublished - 2021

Keywords

  • Bivariant theory
  • Equivariant localization
  • Riemann-Roch theorems

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