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arXiv · 1907.00076

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

Dave Anderson·Richard Gonzales·Sam Payne
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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.

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Dave Anderson, Richard Gonzales, Sam Payne. 2019-06-28. Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory. https://doi.org/10.2140/ant.2021.15.341

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