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

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.

arXiv comment46 pages, with an appendix by G. Vezzosi; v3: minor corrections, to appear in Algebra Number Theory

References in corpus (10)

• Higher and derived stacks: a global overview
• Virtual Cartier divisors and blow-ups
• Bivariant Algebraic Cobordism
• Proper local complete intersection morphisms preserve perfect complexes
• Operational K-theory
• Precobordism and cobordism
• Homogeneous spaces and equivariant embeddings
• Equivariant $K$-theory of smooth projective spherical varieties
• Computing torus-equivariant K-theory of singular varieties
• The cone theorem and the vanishing of Chow cohomology

Cited by in corpus (3)

• Equivariant K-theory classes of matrix orbit closures
• Co-operational bivariant theory
• Minkowski weights and the Grothendieck group of a toric variety