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Spectral weight filtrations

Peter J. Haine, Piotr Pstrągowski

TL;DR

The paper develops a weight-filtration framework for motivic homotopy theory by leveraging a new description of the motivic category away from the characteristic as sheaves on Pure motives, enabling canonical filtrations on Betti and étale realizations with complex-orientable coefficients. It proves that these filtrations satisfy $\\ell\mathrm{dh}$-descent, providing effective computational methods in positive characteristic via ldH hypercovers, and shows that in the complex orientable case the filtrations lift to stable homotopy types through synthetic realizations. The Gillet–Soulé filtration is recovered as a descent-compatible specialization of the Betti filtration on compactly supported cochains, and the framework connects to Bondarko’s weight structures while enriching them with homotopy-coherent, functorial filtrations across realizations. The construction of filtered Betti and étale realizations, together with the synthetic Betti realization, yields a robust, functorial, and compatible approach to weight data in motivic contexts with wide applicability to cohomology theories and to characteristic $p$ phenomena.

Abstract

We provide a description of Voevodsky's $\infty$-category of motivic spectra in terms of the subcategory of motives of smooth proper varieties. As applications, we construct weight filtrations on the Betti and étale cohomologies of algebraic varieties with coefficients in any complex oriented ring spectrum. We show that these filtrations satisfy $\ell\mathrm{dh}$-descent, giving an effective way of calculating them in positive characteristic. In the complex motivic case, we further refine the weight filtration to one defined at the level of stable homotopy types.

Spectral weight filtrations

TL;DR

The paper develops a weight-filtration framework for motivic homotopy theory by leveraging a new description of the motivic category away from the characteristic as sheaves on Pure motives, enabling canonical filtrations on Betti and étale realizations with complex-orientable coefficients. It proves that these filtrations satisfy -descent, providing effective computational methods in positive characteristic via ldH hypercovers, and shows that in the complex orientable case the filtrations lift to stable homotopy types through synthetic realizations. The Gillet–Soulé filtration is recovered as a descent-compatible specialization of the Betti filtration on compactly supported cochains, and the framework connects to Bondarko’s weight structures while enriching them with homotopy-coherent, functorial filtrations across realizations. The construction of filtered Betti and étale realizations, together with the synthetic Betti realization, yields a robust, functorial, and compatible approach to weight data in motivic contexts with wide applicability to cohomology theories and to characteristic phenomena.

Abstract

We provide a description of Voevodsky's -category of motivic spectra in terms of the subcategory of motives of smooth proper varieties. As applications, we construct weight filtrations on the Betti and étale cohomologies of algebraic varieties with coefficients in any complex oriented ring spectrum. We show that these filtrations satisfy -descent, giving an effective way of calculating them in positive characteristic. In the complex motivic case, we further refine the weight filtration to one defined at the level of stable homotopy types.
Paper Structure (36 sections, 49 theorems, 278 equations)

This paper contains 36 sections, 49 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. Motivation and overview
  3. The complex orientable case
  4. A new description of motivic spectra
  5. Filtered étale realization
  6. Descent and the Gillet--Soulé filtration
  7. Synthetic Betti realization
  8. Relation to Bondarko's work on weight structures
  9. Recollections on motivic homotopy theory
  10. Motivic spectra and the six operations
  11. Compactly supported motives of schemes
  12. Betti realization
  13. Étale realization
  14. Motivic spectra as sheaves on pure motives
  15. Perfect pure motivic spectra
  16. ...and 21 more sections

Key Result

Theorem 1.2.1

Let $A \in \mathop{\mathrm{CAlg}}\nolimits(\mathop{\mathrm{Sp}}\nolimits)$ be complex orientable. Then, there exists a unique colimit-preserving lax symmetric monoidal functor such that on the subcategory of motivic spectra of the form $S \simeq (\mathbb{P}^{1})^{\otimes n} \otimes \Sigma_{+}^{\infty} Y$ with $n \in \mathbb{Z}$ and $Y$ a smooth proper complex variety, we have a natural equivalenc

Theorems & Definitions (146)

  • Theorem 1.2.1: \ref{['cor:homological_filtered_Betti_realization_exists_for_complex_orientable_rings']}
  • Theorem 1.3.1: \ref{['theorem:shk_is_additive_sheaves_on_perfect_pure_motives']}
  • Remark 1.3.2: inverting $e$
  • Corollary 1.3.4
  • Theorem 1.4.1: \ref{['proposition:existence_of_the_a_linear_etale_realization']}
  • Theorem 1.5.1: \ref{['theorem:hypercovers_of_varieties_are_colimit_diagrams_in_shk']}
  • Theorem 1.5.2: \ref{['theorem:weights_on_bm_homology_of_proper_variety_calculated_through_an_ldh_hypercover']}
  • Theorem 1.5.4: \ref{['theorem:gillet_soule_filtration_comparison']}
  • Remark 1.5.6
  • Theorem 1.6.2: \ref{['theorem:existence_of_complex_synthetic_betti_homology']}
  • ...and 136 more