TY - JOUR

T1 - Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory

AU - Anderson, Dave

AU - Gonzales, Richard

AU - Payne, Sam

AU - Vezzosi, Gabriele

N1 - Publisher Copyright:
© 2021, Mathematical Science Publishers. All rights reserved.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Bivariant theory

KW - Equivariant localization

KW - Riemann-Roch theorems

UR - http://www.scopus.com/inward/record.url?scp=85105209132&partnerID=8YFLogxK

U2 - 10.2140/ant.2021.15.341

DO - 10.2140/ant.2021.15.341

M3 - Article

AN - SCOPUS:85105209132

SN - 1937-0652

VL - 15

SP - 341

EP - 385

JO - Algebra and Number Theory

JF - Algebra and Number Theory

IS - 2

ER -