On the Motivic Homotopy Type of Algebraic Stacks

Technical Summary: On the Motivic Homotopy Type of Algebraic Stacks

Problem Statement
The paper addresses the challenge of extending motivic homotopy theory and the theory of motives from schemes to general algebraic stacks. While techniques such as homotopical descent allow for the reduction of stack problems to the scheme case, this reduction has historically been limited to specific classes of stacks, notably those that are Nisnevich locally quotient stacks (e.g., stacks with linearly reductive stabilizers). A significant gap existed in establishing that the standard formalisms of motives and motivic homotopy theory yield "correct" results for arbitrary algebraic stacks, particularly those lacking local quotient structures or specific separation hypotheses. Furthermore, there was a need to unify different definitions of the stable motivic homotopy category for stacks, which previously required the existence of smooth presentations with Nisnevich local sections.

Methodology
The authors develop a framework based on smooth-Nisnevich coverings. A morphism $f: Y \to X$ of algebraic stacks is defined as a smooth-Nisnevich covering if it is smooth and every morphism from the spectrum of a field $\text{Spec } K \to X$ lifts to $Y$.

The core methodological contribution is the construction of such coverings for any algebraic stack, removing previous constraints regarding separation, finite type, or stabilizer properties. The authors utilize:

1. Splitting Covers: They establish that a smooth morphism admits a splitting cover (a finite partition of the base where the morphism admits sections) if and only if it is a smooth-Nisnevich covering.
2. Hilbert Stacks: The proof of existence relies on the properties of Hilbert stacks, specifically the substacks parameterizing quasi-finite or étale families. By analyzing the Hilbert stack of a smooth cover, they construct a new cover that is representable and possesses the desired lifting properties.
3. Čech Nerves: Once a smooth-Nisnevich covering $p: \mathcal{X} \to X$ is established (where $\mathcal{X}$ is a scheme), the authors analyze the associated Čech nerve $\mathcal{X}_\bullet$. They prove that this simplicial scheme induces an equivalence in the Morel-Voevodsky $\mathbb{A}^1$-homotopy category.

Key Contributions and Results

• Theorem 1.2 (Existence of Smooth-Nisnevich Covers): The paper proves that for any algebraic stack $X$, there exists a smooth-Nisnevich covering $p: \mathcal{X} \to X$ where $\mathcal{X}$ is a scheme.

  • If $X$ is quasi-compact and quasi-separated (qcqs) with affine stabilizers, $\mathcal{X}$ can be chosen to be affine.
  • If $X$ is a Deligne–Mumford stack, $p$ can be chosen to be étale.
  • This result generalizes previous findings by eliminating separation hypotheses and noetherian assumptions required in earlier literature (e.g., [LMB00], [Čes15], [Pir18]).

• Theorem 1.3 (Homotopy Descent): Let $p: \mathcal{X} \to X$ be a smooth-Nisnevich covering over a field $k$, and let $\mathcal{X}_\bullet$ be its Čech nerve. The morphism $\mathcal{X}_\bullet \to X$ induces an equivalence in the Morel-Voevodsky $\mathbb{A}^1$-homotopy category, $\mathcal{H}(k)$.

  • This establishes that the Nisnevich homotopy type of any algebraic stack can be described by a simplicial scheme, making homotopical descent arguments accessible for all stacks.

• Motivic Properties of Smooth Stacks (Theorem 3.4): As a consequence of the descent result, the motive $M(X)$ of a smooth algebraic stack $X$ satisfies the same fundamental properties as the motive of a smooth scheme:

  • Nisnevich descent.
  • Projective bundle formula.
  • Blow-up formula.
  • Gysin triangle.
  • These results generalize those in [CDH20], which were previously restricted to Nisnevich locally quotient stacks.

• Unification of Stable Homotopy Categories (Corollary 3.13): The paper demonstrates that for any quasi-separated algebraic stack $X$, the limit-extended stable motivic homotopy category $\mathcal{SH}^\wedge(X)$ (from [KR21]) and the lisse-extended category $\mathcal{SH}^{\otimes}_{ext}(X)$ (from [Cho21]) are equivalent. This equivalence holds because Theorem 1.2 guarantees that every stack admits a smooth presentation with Nisnevich local sections.

• Poincaré Duality for Motives with Compact Support (Theorem 3.10): For a smooth algebraic stack $X$ of dimension $d$ with affine stabilizers, the authors define the motive with compact support $M^c(X)$ via a homotopy limit over a smooth-Nisnevich cover. They prove the isomorphism $M^c(X) \cong M(X)^\vee(d)[2d]$.

Significance
The paper asserts that the existence of smooth-Nisnevich coverings for arbitrary algebraic stacks is a foundational result that validates the application of motivic homotopy theory to the general stack setting. By showing that the motive of a smooth stack behaves analogously to that of a smooth scheme, the authors confirm that the various formalisms of motives produce consistent results without requiring the restrictive "locally quotient" condition.

Additionally, the paper notes that the boundedness result in Theorem 1.2 has applications beyond motivic theory, specifically in applying Elkik's approximation technique to the Picard stack to study Grothendieck's question regarding the injectivity of the map $H^2(X, \mathbb{G}_m) \to \varprojlim H^2(X_n, \mathbb{G}_m)$ for proper morphisms over complete Noetherian rings.

The authors acknowledge that a similar existence result was recently proved in [CD24], but emphasize that their contribution includes the specific boundedness result for qcqs stacks with affine stabilizers and the explicit application to the unification of stable motivic homotopy categories and motivic cohomology properties.