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Quadratic Riemann-Roch formulas

Frédéric Déglise, Jean Fasel

Abstract

In this article, we produce Grothendieck-Riemann-Roch formulas for cohomology theories that are not oriented in the classical sense. We then specialize to the case of cohomology theories that admit a so-called symplectic orientation and show how to compute the relevant Todd classes in that situation. At the end of the article, we illustrate our methods on the Borel character linking Hermitian K-theory and rational MW-motivic cohomology.

Quadratic Riemann-Roch formulas

Abstract

In this article, we produce Grothendieck-Riemann-Roch formulas for cohomology theories that are not oriented in the classical sense. We then specialize to the case of cohomology theories that admit a so-called symplectic orientation and show how to compute the relevant Todd classes in that situation. At the end of the article, we illustrate our methods on the Borel character linking Hermitian K-theory and rational MW-motivic cohomology.
Paper Structure (20 sections, 40 theorems, 213 equations)

This paper contains 20 sections, 40 theorems, 213 equations.

Table of Contents

  1. Plan of the article
  2. Orientation theory and fundamental classes
  3. Recollections on bivariant theories
  4. Orientations and associated virtual Thom classes
  5. Euler classes
  6. Fibred categories and virtual Thom classes
  7. Fundamental classes and Grothendieck-Riemann-Roch formulas
  8. Oriented fundamental classes
  9. Todd classes and GRR formulas
  10. Symplectic orientations and formal ternary laws
  11. Recollections on FTL
  12. $\mathrm{Sp}$-oriented theories
  13. Virtual symplectic Thom classes
  14. Higher Borel classes and $\mathop{\mathrm{GW}}\nolimits$-linearity
  15. The FTL associated to an $\mathrm{Sp}$-oriented spectrum
  16. ...and 5 more sections

Key Result

Proposition 2

(See def:orientedfundamentalclass) Let $\mathbb E$ be a $\sigma$-oriented ring spectrum, and let $f:X \rightarrow S$ be a smoothable lci morphism. A $\sigma$-orientation of $f$ is a class $\tilde{\tau}_f \in \mathop{\mathrm{\underline K}}\nolimits^\sigma(X)$ such that $\sigma(\tilde{\tau}_f)$ is iso where $\mathbb E_{n,i}(X/S)=\mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathrm{SH}}\nolimits(S)}(\math

Theorems & Definitions (107)

  • Definition 1
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6: quadratic HRR formula
  • Definition 1.1.2
  • Remark 1.1.3
  • Proposition 1.1.4
  • Remark 1.1.5
  • ...and 97 more