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A motivic spectrum representing hermitian K-theory

Baptiste Calmès, Yonatan Harpaz, Denis Nardin

TL;DR

This work removes the standard restriction that 2 be invertible to develop motivic hermitian K-theory, establishing Nisnevich descent for both quadratic and symmetric Grothendieck–Witt theories and proving a projective bundle formula, dévissage, and A^1-invariance on regular Noetherian bases of finite Krull dimension. It introduces and exploits the framework of Poincaré ∞-categories and Karoubi-localising approximations to represent symmetric GW-theory by a motivic E∞-ring spectrum KQ_S and related motivic spectra KW_S and KGL_S, with base-change invariance ensuring a canonical pullback to a universal object KQ. The results yield a robust motivic realization functor linking GW/L/K-theory to motivic spectra, clarifying the multiplicative structure and relations among these invariants (e.g., Tate and Wood sequences) and enabling a unified approach to both GW-theory and its Karoubi-localising variants in the motivic setting. Overall, the paper advance the motivic theory of hermitian K-theory beyond 2-invertibility, providing descent, multiplicativity, and invariant properties essential for further arithmetic and geometric applications in motivic homotopy theory.

Abstract

We establish fundamental motivic results about hermitian K-theory without assuming that 2 is invertible on the base scheme. In particular, we prove that both quadratic and symmetric Grothendieck-Witt theory satisfy Nisnevich descent, and that symmetric Grothendieck-Witt theory further satisfies a projective bundle formula, as well as dévissage and A^1-invariance over a regular Noetherian base of finite Krull dimension. We use this to show that over a regular Noetherian base, symmetric Grothendieck-Witt theory is represented by a motivic E-infinity-ring spectrum, which we then show is an absolutely pure spectrum, answering a question of Déglise. As with algebraic K-theory, we show that over a general base, one can also construct a hermitian K-theory motivic spectrum, representing this time a suitable homotopy invariant and Karoubi-localising version of Grothendieck-Witt theory.

A motivic spectrum representing hermitian K-theory

TL;DR

This work removes the standard restriction that 2 be invertible to develop motivic hermitian K-theory, establishing Nisnevich descent for both quadratic and symmetric Grothendieck–Witt theories and proving a projective bundle formula, dévissage, and A^1-invariance on regular Noetherian bases of finite Krull dimension. It introduces and exploits the framework of Poincaré ∞-categories and Karoubi-localising approximations to represent symmetric GW-theory by a motivic E∞-ring spectrum KQ_S and related motivic spectra KW_S and KGL_S, with base-change invariance ensuring a canonical pullback to a universal object KQ. The results yield a robust motivic realization functor linking GW/L/K-theory to motivic spectra, clarifying the multiplicative structure and relations among these invariants (e.g., Tate and Wood sequences) and enabling a unified approach to both GW-theory and its Karoubi-localising variants in the motivic setting. Overall, the paper advance the motivic theory of hermitian K-theory beyond 2-invertibility, providing descent, multiplicativity, and invariant properties essential for further arithmetic and geometric applications in motivic homotopy theory.

Abstract

We establish fundamental motivic results about hermitian K-theory without assuming that 2 is invertible on the base scheme. In particular, we prove that both quadratic and symmetric Grothendieck-Witt theory satisfy Nisnevich descent, and that symmetric Grothendieck-Witt theory further satisfies a projective bundle formula, as well as dévissage and A^1-invariance over a regular Noetherian base of finite Krull dimension. We use this to show that over a regular Noetherian base, symmetric Grothendieck-Witt theory is represented by a motivic E-infinity-ring spectrum, which we then show is an absolutely pure spectrum, answering a question of Déglise. As with algebraic K-theory, we show that over a general base, one can also construct a hermitian K-theory motivic spectrum, representing this time a suitable homotopy invariant and Karoubi-localising version of Grothendieck-Witt theory.
Paper Structure (49 sections, 112 theorems, 528 equations)

This paper contains 49 sections, 112 theorems, 528 equations.

Table of Contents

  1. Introduction
  2. Recollections on hermitian K-theory
  3. Poincaré $\infty$-categories
  4. Hermitian K-theory
  5. Karoubi-Grothendieck-Witt theory
  6. Variants of Poincaré structures
  7. Poincaré structures on derived categories of schemes
  8. Multiplicative structures on hermitian and bilinear categories
  9. Poincaré categories associated to rigid symmetric monoidal structures
  10. Derived Poincaré categories of schemes
  11. Genuine structures
  12. Nisnevich descent
  13. Flat linear categories
  14. Bounded Karoubi projections
  15. Zariski descent
  16. ...and 34 more sections

Key Result

Theorem 1.0.1

Let $S$ be a regular Noetherian scheme of finite Krull dimension. Then symmetric Grothendieck-Witt theory is represented by a motivic spectrum over $S$.

Theorems & Definitions (271)

  • Theorem 1.0.1
  • Proposition 1.0.2: Nisnevich descent
  • Proposition 1.0.3: Projective line formula
  • Proposition 1.0.4: Bott periodicity
  • Theorem 1.0.5
  • Theorem 1.0.6: Devissage
  • Proposition 1.0.7: Base change invariance
  • Proposition 1.0.8
  • Example 2.1.1
  • Definition 3.0.1
  • ...and 261 more