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On the Motivic Homotopy Type of Algebraic Stacks (with an appendix joint with Jack Hall)

Neeraj Deshmukh, Jack Hall

TL;DR

The paper constructs a Nisnevich-based motive theory for algebraic stacks by proving that any smooth stack admits a smooth-Nisnevich presentation whose Čech nerve models its ${\mathbb A}^1$-homotopy type, thereby enabling descent from schemes to stacks. It defines the motive $M(\mathcal{X})$ of a stack in ${\mathbf DM}^{\mathrm{eff}}(k,{\mathbb Z})$, and shows that fundamental motivic properties such as descent, projective bundle formula, blow-up formulas, and Gysin triangles extend from schemes to stacks via GL$_n$-presentations. The work develops motivic cohomology for stacks, identifying its computation with hypercohomology on the smooth-Nisnevich site and giving an Ext–Hom interpretation; it also proves a Nisnevich/Étale comparison with Chow groups for quotient stacks and establishes compactly supported motives with Poincaré duality. Finally, it analyzes the stable motivic homotopy category of stacks, showing equivalences between competing definitions under smooth-Nisnevich coverage and outlining connections to framed motivic spectra. Overall, the framework unifies motivic homotopy and cohomology theories for stacks with that of schemes, enabling robust computational and structural results for a broad class of stacks.

Abstract

We construct smooth presentations of algebraic stacks that are local epimorphisms in the Morel-Voevodsky $\mathbb{A}^1$-homotopy category. As a consequence we show that the motive of a smooth stack (in Voevodsky's triangulated category of motives) has many of the same properties as the motive of a smooth scheme.

On the Motivic Homotopy Type of Algebraic Stacks (with an appendix joint with Jack Hall)

TL;DR

The paper constructs a Nisnevich-based motive theory for algebraic stacks by proving that any smooth stack admits a smooth-Nisnevich presentation whose Čech nerve models its -homotopy type, thereby enabling descent from schemes to stacks. It defines the motive of a stack in , and shows that fundamental motivic properties such as descent, projective bundle formula, blow-up formulas, and Gysin triangles extend from schemes to stacks via GL-presentations. The work develops motivic cohomology for stacks, identifying its computation with hypercohomology on the smooth-Nisnevich site and giving an Ext–Hom interpretation; it also proves a Nisnevich/Étale comparison with Chow groups for quotient stacks and establishes compactly supported motives with Poincaré duality. Finally, it analyzes the stable motivic homotopy category of stacks, showing equivalences between competing definitions under smooth-Nisnevich coverage and outlining connections to framed motivic spectra. Overall, the framework unifies motivic homotopy and cohomology theories for stacks with that of schemes, enabling robust computational and structural results for a broad class of stacks.

Abstract

We construct smooth presentations of algebraic stacks that are local epimorphisms in the Morel-Voevodsky -homotopy category. As a consequence we show that the motive of a smooth stack (in Voevodsky's triangulated category of motives) has many of the same properties as the motive of a smooth scheme.
Paper Structure (8 sections, 14 theorems, 8 equations)

This paper contains 8 sections, 14 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. The motive of an algebraic stack
  3. Motivic cohomology
  4. Hypercohomology
  5. Motivic cohomology with finite coefficients
  6. Motives with compact support
  7. The stable motivic homotopy category of a stack
  8. Smooth-Nisnevich covers (joint with Jack Hall)

Key Result

Theorem 1.2

Let $p:X\rightarrow {\mathcal{X}}$ be a smooth-Nisnevich covering over a field $k$. Let $X_{\bullet}$ denote the Čech nerve of $p$. Then the morphism $p_{\bullet}:X_{\bullet}\rightarrow {\mathcal{X}}$ induces an equivalence in the Morel-Voevodsky ${\mathbb A}^1$-homotopy category, ${\mathcal{H}}(k)$

Theorems & Definitions (39)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • proof : Proof of Theorem \ref{['corollary-smooth-nisnevich-presentation']}
  • Corollary 2.3
  • proof
  • ...and 29 more