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Weight Filtrations and Derived Motivic Measures

Anubhav Nanavaty

TL;DR

The work constructs derived motivic measures by lifting classical motivic invariants to spectrum-level maps from $K( ext{Var}_k)$, leveraging a generalized Gillet--Soulé weight complex encoded as pro-weight complexes in a Waldhausen $K$-theory framework. It develops a Waldhausen structure on simplicial smooth projective varieties, proves equivalences for pro-objects, and builds a pro-Weight Complex category to capture all construction choices; these culminate in a derived measure to simplicial $cdh$-sheaves and recover the derived Voevodsky and Chow-motive measures. The results yield a well-defined Gillet--Soulé weight filtration in derived settings, extend to stable/unstable homotopy types, and produce a derived compactly supported $\\mathbb{A}^1$ Euler characteristic landing in $ ext{GW}(k)$ via $ ext{End}(1_{ ext{SH}_k})$. Altogether, the paper provides a coherent higher-categorical and $K$-theoretic framework linking Chow motives, Voevodsky motives, and motivic measures, with potential impacts on understanding higher $K$-groups of varieties and their invariants.

Abstract

Let $k$ be a field admitting resolution of singularities. We lift a number of motivic measures, such as the Gillet-Soulé measure and the compactly supported $\mathbb{A}^1$-Euler characteristic, to derived motivic measures in the sense of Campbell-Wolfson-Zakharevich, answering various questions in the literature. We do so by generalizing the construction of the Gillet-Soulé weight complex to show that it is well-defined up to a certain notion of weak equivalence in the category of simplicial smooth projective varieties. For a $k$-variety $X$, the collection of all Gillet-Soulé weight complexes of $X$ form a 'weakly constant' pro-object of simplicial varieties, and under mild assumptions, the $K$-theory of a Waldhausen category is equivalent to the $K$-theory of its weakly constant pro-objects. This leads us to a new proof of the existence of the Gillet-Soulé weight filtration, along with the weight filtration on both the stable and unstable homotopy type of a variety over $k$. We show these constructions provide the aforementioned derived motivic measures, or maps of spectra, out of $K(Var_k)$, the Zakharevich $K$-theory of varieties.

Weight Filtrations and Derived Motivic Measures

TL;DR

The work constructs derived motivic measures by lifting classical motivic invariants to spectrum-level maps from , leveraging a generalized Gillet--Soulé weight complex encoded as pro-weight complexes in a Waldhausen -theory framework. It develops a Waldhausen structure on simplicial smooth projective varieties, proves equivalences for pro-objects, and builds a pro-Weight Complex category to capture all construction choices; these culminate in a derived measure to simplicial -sheaves and recover the derived Voevodsky and Chow-motive measures. The results yield a well-defined Gillet--Soulé weight filtration in derived settings, extend to stable/unstable homotopy types, and produce a derived compactly supported Euler characteristic landing in via . Altogether, the paper provides a coherent higher-categorical and -theoretic framework linking Chow motives, Voevodsky motives, and motivic measures, with potential impacts on understanding higher -groups of varieties and their invariants.

Abstract

Let be a field admitting resolution of singularities. We lift a number of motivic measures, such as the Gillet-Soulé measure and the compactly supported -Euler characteristic, to derived motivic measures in the sense of Campbell-Wolfson-Zakharevich, answering various questions in the literature. We do so by generalizing the construction of the Gillet-Soulé weight complex to show that it is well-defined up to a certain notion of weak equivalence in the category of simplicial smooth projective varieties. For a -variety , the collection of all Gillet-Soulé weight complexes of form a 'weakly constant' pro-object of simplicial varieties, and under mild assumptions, the -theory of a Waldhausen category is equivalent to the -theory of its weakly constant pro-objects. This leads us to a new proof of the existence of the Gillet-Soulé weight filtration, along with the weight filtration on both the stable and unstable homotopy type of a variety over . We show these constructions provide the aforementioned derived motivic measures, or maps of spectra, out of , the Zakharevich -theory of varieties.
Paper Structure (14 sections, 75 theorems, 143 equations)

This paper contains 14 sections, 75 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Preliminaries on The Chow Weight Complex, Pro-objects, and K-theory
  4. Chow Motives and the Gillet--Soulé Construction
  5. Waldhausen $K$-theory
  6. Pro-objects
  7. A Waldhausen Category of Simplicial Smooth Projective Varieties
  8. Pro-Objects of Waldhausen Categories
  9. The Category of Pro-Weight Complexes
  10. Lifting the Measure to $\mathrm{s}\mathbb{V}_*^{\mathrm{perf}}$
  11. The Derived Measure To Simplicial cdh Sheaves
  12. Recovering the Derived Voevodsky Measure
  13. From Voevodsky to Chow Motives
  14. The Derived Compactly Supported $\mathbb{A}^1$ Euler Characteristic

Key Result

Theorem 1.1

Let $k$ be a field admitting resolution of singularities. There exists a homotopy commutative diagram of spectra \begin{tikzcd} &&K(\sShv_{\cdh}^{\perf}(\Sch(k))_*\arrow[r,"\ref{prop:symspt}"]\arrow[d,"\ref{thm:doldkan}"]&K(\mathrm{SH}^{\perf}(k))\arrow[r,"\ref{cor:trace}"]&\End(1_{\mathrm{SH}(k

Theorems & Definitions (177)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Theorems \ref{['thm:DKequiv']} and \ref{['thm:proobj']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: 1.4.1 of GS
  • ...and 167 more