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Power structures on the Grothendieck--Witt ring and the motivic Euler characteristic

Jesse Pajwani, Ambrus Pál

Abstract

For $k$ a field, we construct a power structure on the Grothendieck--Witt ring of $k$ which has the potential to be compatible with symmetric powers of varieties and the motivic Euler characteristic. We then show this power structure is compatible with the power structure when we restrict to varieties of dimension $0$.

Power structures on the Grothendieck--Witt ring and the motivic Euler characteristic

Abstract

For a field, we construct a power structure on the Grothendieck--Witt ring of which has the potential to be compatible with symmetric powers of varieties and the motivic Euler characteristic. We then show this power structure is compatible with the power structure when we restrict to varieties of dimension .
Paper Structure (13 sections, 61 theorems, 84 equations)

This paper contains 13 sections, 61 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Acknowledgements
  4. Power structures on rings
  5. Determining a possible power structure on $\widehat{\mathrm{W}}(k)$
  6. Uniqueness of the power structure
  7. Determining the power structure
  8. The power structure is well defined
  9. Properties of this power structure
  10. Compatibility in dimension $0$
  11. Quadratic and biquadratic étale algebras
  12. Twisting results for the power structure
  13. From multiquadratic algebras to all étale algebras

Key Result

Theorem 1.2

Define $t_\alpha$ to be the element of $\widehat{\mathrm{W}}(k)$ given by $\langle 2 \rangle + \langle \alpha \rangle - \langle 1 \rangle - \langle 2\alpha\rangle$. Then there is a unique power structure on $\widehat{\mathrm{W}}(k)$ such that for any $\alpha \in k^\times$

Theorems & Definitions (133)

  • Definition 1.1
  • Theorem 1.2: Corollary $\ref{['finalpstruc']}$
  • Theorem 1.3: Corollary $\ref{['maincor1']}$
  • Theorem 1.4: Corollary $\ref{['maincor2']}$
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3: Proposition 1 of GZLMH
  • Corollary 2.4
  • proof
  • ...and 123 more