Profinite Borel completeness and smooth Artin motives

Technical Summary: Profiniteness, Borel Completeness, and Smooth Artin Motives

Problem Statement
The paper addresses the relationship between two distinct mathematical frameworks: the $\infty$-category of smooth Artin motives over a general base scheme $S$ and the representation theory of the étale fundamental group $\pi_1^{\text{ét}}(S)$. While Voevodsky established an equivalence between Artin motives over a field $F$ and derived permutation modules of the absolute Galois group $\text{Gal}(F^{\text{sep}}/F)$, it remained an open question whether this equivalence extends to general base schemes $S$ where the Galois group is replaced by the profinite étale fundamental group $\pi_1^{\text{ét}}(S)$.

A secondary, foundational problem arises in the equivariant homotopy theory of profinite groups. For finite groups, the subcategory of Borel complete spectra is well-understood as a functor category (via Mathew–Naumann–Noel). However, for profinite groups, the orbit $G/1$ is not a finite $G$-set, complicating the direct generalization of Borel completeness. The paper investigates whether there are multiple, distinct notions of "Borel completeness" for profinite groups and how they relate to sheaf theory and motives.

Methodology
The author employs a two-pronged approach, first developing the necessary equivariant homotopy theory and then applying it to the theory of motives.

1. Equivariant Homotopy Theory for Profinite Groups:

   • The paper revisits the description of Borel complete spectra for finite groups, introducing coefficients in a connective commutative ring spectrum $R$.
   • For a profinite group $G = \lim_i G_i$, two notions of Borel completeness are defined:
     • Levelwise Borel completeness ($Sp^{\text{lwBorel}}_G$): Defined as the limit of Borel complete categories of the finite quotients $G_i$ along categorical fixed points.
     • Borel completeness ($Sp^{\text{Borel}}_G$): Defined via completeness with respect to a specific algebra object $A$, constructed as the colimit of the duals of orbits of all open normal subgroups.
   • The author utilizes spectral Mackey functors and the theory of module categories over equivariant ring spectra to analyze these structures. A key technical tool is the identification of the relationship between these two notions via $t$-structures and hypercompletion.

2. Motives and Correspondences:

   • The paper constructs the $\infty$-category of effective Voevodsky motives $DM^{\text{eff}}_{\tau}(S; R)$ (for $\tau = \text{Nisnevich}$ or $\text{étale}$) using finite correspondences with coefficients.
   • Smooth Artin motives $DAM_{\tau}(S; R)$ are defined as the localising subcategory generated by motives of finite étale $S$-schemes.
   • The author establishes a bridge between these motives and the equivariant categories defined in the first part by utilizing Grothendieck's equivalence between finite étale $S$-schemes and finite discrete $G$-sets (where $G = \pi_1^{\text{ét}}(S)$).
   • Crucially, the paper investigates the behavior of "transfers" (finite correspondences) in the context of étale sheaves on finite étale schemes, proving that étale sheaves on these schemes naturally admit transfers.

Key Contributions and Results

• Distinction of Borel Completeness: The paper identifies that for profinite groups, "levelwise" Borel completeness and the standard "Borel completeness" (defined via the algebra $A$) are generally distinct.

  • Theorem 3.30: The category of Borel complete $G$-spectra is equivalent to the hypercompletion of the category of levelwise Borel complete $G$-spectra. This provides a precise structural relationship between the two notions.

• Equivalence with Sheaves and Representations:

  • Proposition 4.8 & 4.13: The paper establishes a commutative diagram connecting the Borel complete categories of equivariant spectra to categories of sheaves on finite $G$-sets and derived categories of discrete modules.
  • Specifically, the category of levelwise Borel complete modules is equivalent to the category of sheaves on finite $G$-sets, while the Borel complete category corresponds to hypersheaves (under specific conditions on the virtual cohomological dimension of $G$).

• Generalization of Voevodsky's Theorem (Nisnevich Case):

  • Theorem 5.19: For a connected, noetherian scheme $S$ of finite Krull dimension, there is a symmetric monoidal equivalence between the $\infty$-category of smooth Artin motives (Nisnevich topology) and the derived category of permutation modules over the profinite group $\pi_1^{\text{ét}}(S)$:
    $$DAM^{\text{Nis}}(S; R) \simeq D\text{Perm}(\pi_1^{\text{ét}}(S, s); R)$$
  • This result confirms that the classical description of Artin motives via Galois representations extends to general base schemes in the Nisnevich topology.

• Étale Case and Transfers:

  • Theorem 5.34: The paper proves that étale sheaves on finite étale $S$-schemes automatically possess transfers. This identifies the category of étale sheaves with transfers on finite étale schemes with the category of étale sheaves on finite étale schemes (without explicit reference to correspondences).
  • Theorem 5.37: A comprehensive commutative diagram is presented, linking derived permutation modules, Borel complete spectra, and smooth Artin motives for both Nisnevich and étale topologies.
  • Corollary 5.31: In the hypercomplete étale case, under assumptions that the coefficient ring $R$ has positive characteristic $n$ and residue characteristics of $S$ are prime to $n$, the category of smooth Artin motives is equivalent to the category of hypersheaves with transfers on finite étale schemes.

Significance and Claims
The paper claims to resolve the question of whether Voevodsky's description of Artin motives holds for general base schemes. It affirms this for the Nisnevich topology and provides a nuanced answer for the étale topology, highlighting the role of hypercompletion.

The significance lies in:

1. Unifying Framework: It unifies the theory of motives with equivariant homotopy theory for profinite groups, showing that smooth Artin motives are controlled by the étale fundamental group.
2. Clarification of Profinite Equivariance: It clarifies the subtle distinction between levelwise and hypercomplete Borel completeness in the profinite setting, identifying the latter as the correct notion for capturing the full structure of étale sheaves (hypersheaves).
3. Transfer of Transfers: It provides an independent proof that étale sheaves on finite étale schemes inherently carry transfers, a result previously known but reliant on specific hom-set identifications in other literature.

The author notes that while the Nisnevich case yields a clean equivalence, the étale case requires distinguishing between sheaves and hypersheaves, and the equivalence to permutation modules in the non-hypercomplete étale case requires further computation of mapping spaces which is not performed in this work. The results are presented as a foundational step, relying on recent work by Balmer–Gallauer and Dubey–Gómez regarding the Balmer spectra of these categories.