Pro-étale motives and solid rigidity

1. Problem Statement and Motivation

The paper addresses fundamental limitations in the current frameworks for $\ell$-adic cohomology and motivic theory, specifically concerning the interaction between topological coefficients (like $\mathbb{Z}_\ell$ and $\mathbb{Q}_\ell$) and functoriality (the six operations).

• The $\ell$-adic Dilemma: Classical theories of constructible $\ell$-adic sheaves (e.g., Ekedahl's theory) are constructed via $\ell$-completion of derived categories of étale sheaves. A major flaw is that this process destroys rationalization: $D((-)_{\text{ét}}, \mathbb{Z})^\wedge_\ell \otimes \mathbb{Q} \simeq 0$. Conversely, using the pro-étale site (as in Bhatt-Scholze) allows for a natural definition of $\mathbb{Z}_\ell$ and $\mathbb{Q}_\ell$ sheaves where rationalization works ($D((-)_{\text{pro-ét}}, \mathbb{Z}_\ell) \otimes \mathbb{Q} \simeq D((-)_{\text{pro-ét}}, \mathbb{Q}_\ell)$), but this category fails to support the full six operations formalism (specifically, proper base change fails).
• Motivic Realization: The theory of Voevodsky motives ($DM_{\text{ét}}$) provides a universal coefficient system with the six operations. However, the standard $\ell$-adic realization functor maps to Ekedahl's category, inheriting its incompatibility with rationalization. Furthermore, the motive representing $\ell$-adic cohomology in $DM_{\text{ét}}$ (the $\ell$-completed unit) is not stable under base change in mixed characteristic.
• Solid Sheaves on Schemes: While Fargues and Scholze developed a robust theory of solid sheaves in the rigid-analytic setting (providing a "non-archimedean completeness" that supports a 5-functor formalism), this theory does not extend directly to schemes. In the analytic setting, the proper projection formula fails, preventing a full six-functor formalism.

Goal: Construct a coefficient system for schemes that:

1. Supports the full six operations formalism ($f^*, f_*, f_!, f^!, \otimes, \mathcal{H}om$).
2. Is compatible with rationalization (handling $\mathbb{Z}_\ell$ and $\mathbb{Q}_\ell$ topologically).
3. Provides a solid realization functor from Voevodsky motives that extends the classical $\ell$-adic realization.

2. Methodology

The authors employ a synthesis of condensed mathematics, pro-étale topology, and motivic homotopy theory.

A. Solid Sheaves on Schemes (Section 1)

The authors adapt the Fargues-Scholze framework of solid sheaves to the category of schemes.

• Definition: For a qcqs scheme $X$ and a set of primes $P$, they define solid sheaves on the pro-étale site $X_{\text{pro-ét}}$ with coefficients in $\widehat{\mathbb{Z}}_P$.
• Construction: They define a "solidification" process. A sheaf is solid if it satisfies a specific condition regarding maps from cofiltered limits of étale maps. Specifically, for a pro-étale map $j: U \to X$ (limit of étale $j_\alpha$), the natural map $j^\sharp \widehat{\mathbb{Z}}_P \to j^\blacktriangledown \widehat{\mathbb{Z}}_P$ (where the latter is the limit of the pushforwards) induces an equivalence on Hom-sets into the solid sheaf.
• Condensed Category Theory: They utilize the theory of condensed $\infty$-categories (internal to the topos of condensed anima). They define the category of solid sheaves $D(X, \Lambda)^\blacktriangledown$ as the solidification of the derived category of pro-étale sheaves $D(X_{\text{pro-ét}}, \Lambda)$ within the category of condensed categories. This involves tensoring with the condensed category of solid modules.

B. Pro-étale Motives (Section 2)

To establish the six operations for solid sheaves, the authors construct a new category of pro-étale motives, $SH_{\text{pro-ét}}(X)$.

• Weakly Smooth Schemes: They replace the category of smooth schemes (used in classical motives) with weakly smooth schemes ($WSm_X$), defined as finite compositions of smooth and weakly étale morphisms.
• Construction: They define $SH_{\text{pro-ét}}(X)$ as the $P^1$-stabilization of $\mathbb{A}^1$-local hypersheaves of spectra on $WSm_X$ equipped with the pro-étale topology.
• Six Operations: They prove that this category admits a six-functor formalism. A crucial technical hurdle is that the localization theorem (essential for defining $f_!$) only holds for quasi-compact open immersions. Consequently, the exceptional functors ($f_!, f^!$) are defined only for finitely presented morphisms, not arbitrary finite type morphisms.
• Embedding: They prove that for locally étale bounded schemes, the classical étale motivic spectra embed fully faithfully into pro-étale motivic spectra.

C. The Rigidity Theorem (Section 3)

The core theoretical result connects solid sheaves to solid motives.

• Solidification of Motives: They apply the condensed solidification procedure to the category of pro-étale motives to obtain solid motives $DM(X, \Lambda)^\blacktriangledown$.
• Rigidity: They prove that the category of solid sheaves $D(X, \Lambda)^\blacktriangledown$ is equivalent to the category of solid motives $DM(X, \Lambda)^\blacktriangledown$ (specifically, the effective solid motives).
• Mechanism: The proof relies on showing that solid sheaves are $\mathbb{A}^1$-invariant and that the relevant maps (related to Kummer sequences and roots of unity) become equivalences after solidification. This establishes that the "solid" condition forces the necessary homotopy invariance and Tate stability.

3. Key Contributions and Results

1. The Six Operations for Solid Sheaves on Schemes

Theorem A & E: The authors prove that the category of solid sheaves $D(X, \Lambda)^\blacktriangledown$ on a qcqs scheme $X$ (where primes in $P$ are invertible) admits a full six-functor formalism.

• Significance: This resolves the issue present in the rigid-analytic setting (where the proper projection formula fails). On schemes, the six operations are fully compatible.

2. The Solid Realization Functor

Theorem 3.3.7 & 3.3.10: They construct a solid realization functor:
$$\rho^\blacktriangledown: DM_{\text{ét}}(X, R) \to D(X, R \otimes \widehat{\mathbb{Z}}_P)^\blacktriangledown$$
This functor is a morphism of six-functor formalisms.

• Significance: Unlike the classical $\ell$-adic realization, this functor is compatible with rationalization and base change. It maps the unit motive to the solid ring spectrum representing $\ell$-adic cohomology.

3. Equivalence with Classical Realizations

Proposition 3.3.11 & Corollary 3.3.12: The solid realization functor, when restricted to constructible motives and coefficients $\mathbb{Z}_\ell$ or $\mathbb{Q}_\ell$, recovers the classical $\ell$-adic and $\mathbb{Q}_\ell$-adic realization functors.

• Significance: This ensures compatibility with established results while providing a more robust framework (presentable categories) that handles rational coefficients naturally.

4. Embedding Theorem

Theorem 2.6.2: Étale motivic spectra embed fully faithfully into pro-étale motivic spectra for locally étale bounded schemes. This bridges the gap between classical motivic theory and the new pro-étale setting.

4. Significance and Impact

• Unification of Topology and Motives: The paper successfully unifies the topological approach to $\ell$-adic cohomology (via pro-étale sites and condensed mathematics) with the homotopical approach (motives).
• Resolution of Rationalization Issues: By working with solid sheaves, the authors bypass the "zero after rationalization" problem of Ekedahl's theory and the "lack of six operations" problem of naive pro-étale sheaves.
• Foundation for Spectral Nori Motives: The authors hint at future work defining "spectral Nori motives" using these constructions, potentially offering a new perspective on the algebra of motives over $\mathbb{Q}$.
• Technical Advancement: The development of the six-functor formalism for solid sheaves on schemes (as opposed to just analytic spaces) is a significant step forward in the application of condensed mathematics to algebraic geometry. It demonstrates that the "solid" condition is the correct notion of completeness for schemes to support a robust duality theory.

In summary, the paper provides a new, rigorous foundation for $\ell$-adic cohomology and motives that is stable under base change, compatible with rationalization, and equipped with a full set of functorial operations, effectively solving long-standing structural problems in the field.