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Notes on Milnor-Witt K-theory

Frédéric Déglise

TL;DR

This work develops Milnor–Witt K-theory $K^{MW}_*$ for fields of arbitrary characteristic, integrating twists and transfers into a coherent functorial framework. It blends Morel–Hopkins–Milnor viewpoints with Barge–Morel obstruction theory to build Chow–Witt groups via Rost–Schmidt complexes, using differential trace maps to handle inseparable extensions and to define transfers. The main contributions include a complete transfer theory compatible with Kato and Rost–Schmidt, explicit residue and degree formulas (including quadratic multiplicities), and a robust axiomatic MW-premodule structure that underpins Chow–Witt theory and quadratic cycle computations. Together with the appendix on Grothendieck duality and traces, these results provide a foundation for arithmetic Chow–Witt theory and pave the way for further work on Chow–Witt groups with Niels Feld and Fangzhou Jin.

Abstract

These notes develop the foundations of Milnor-Witt K-theory for fields of arbitrary characteristic, without any perfectness assumptions. Extending the work of Morel and Feld, we establish all functorial properties of Milnor-Witt K-theory with careful attention to twists. A main new contribution is the computation of transfers in the general, in particular inseparable, case using Grothendieck (differential) trace maps. The results provide a complete framework for the functoriality axioms of Milnor-Witt modules and form the basis for an upcoming work on Chow-Witt groups with Niels Feld and Fangzhou Jin.

Notes on Milnor-Witt K-theory

TL;DR

This work develops Milnor–Witt K-theory for fields of arbitrary characteristic, integrating twists and transfers into a coherent functorial framework. It blends Morel–Hopkins–Milnor viewpoints with Barge–Morel obstruction theory to build Chow–Witt groups via Rost–Schmidt complexes, using differential trace maps to handle inseparable extensions and to define transfers. The main contributions include a complete transfer theory compatible with Kato and Rost–Schmidt, explicit residue and degree formulas (including quadratic multiplicities), and a robust axiomatic MW-premodule structure that underpins Chow–Witt theory and quadratic cycle computations. Together with the appendix on Grothendieck duality and traces, these results provide a foundation for arithmetic Chow–Witt theory and pave the way for further work on Chow–Witt groups with Niels Feld and Fangzhou Jin.

Abstract

These notes develop the foundations of Milnor-Witt K-theory for fields of arbitrary characteristic, without any perfectness assumptions. Extending the work of Morel and Feld, we establish all functorial properties of Milnor-Witt K-theory with careful attention to twists. A main new contribution is the computation of transfers in the general, in particular inseparable, case using Grothendieck (differential) trace maps. The results provide a complete framework for the functoriality axioms of Milnor-Witt modules and form the basis for an upcoming work on Chow-Witt groups with Niels Feld and Fangzhou Jin.
Paper Structure (32 sections, 46 theorems, 264 equations)

This paper contains 32 sections, 46 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Milnor's K-theory and conjectures
  3. Barge and Morel obstruction theory
  4. Basic aim and scope of these notes
  5. Construction of transfers
  6. Towards Chow-Witt groups
  7. Reading guide
  8. Acknowledgments
  9. Milnor-Witt K-theory and Grothendieck-Witt groups
  10. Grothendieck-Witt groups and symmetric bilinear forms
  11. Definition by generators and relations
  12. Relations with quadratic forms
  13. Twists
  14. Residues
  15. A detour on Chow-Witt groups of Dedekind schemes
  16. ...and 17 more sections

Key Result

Theorem 1

(see thm:PB1) Let $k$ be a field, and $\mathcal{L}$ be an invertible sheaf on the projective line $\mathbb P^1_k$. We let $\mathcal{L}_\infty$ be the restriction of $\mathcal{L}$ over the point at $\infty$ and $\omega_\infty$ be the conormal sheaf of the immersion $i_\infty:\{\infty\} \rightarrow \m Explicitly, the isomorphism is given by pushforward along $i_\infty$.

Theorems & Definitions (164)

  • Theorem
  • Theorem : see \ref{['ex:compute_MW-trace']}(3)
  • Definition 2.1.2
  • Remark 2.1.3
  • Example 2.1.4
  • Example 2.1.5
  • Definition 2.1.6
  • Remark 2.1.7
  • Definition 2.1.8
  • Remark 2.1.9
  • ...and 154 more