Coherent six-functor formalisms: Pro vs Solid

Technical Summary: Notes on Coherent Six-Functor Formalisms: Pro vs Solid

Problem Statement
The paper addresses a foundational gap in the Grothendieck six-functor formalism for coherent sheaves. While the formalism is well-established for $\ell$-adic constructible sheaves in étale cohomology, the case for coherent sheaves remains incomplete. Specifically, for an open immersion $j: U \hookrightarrow X$, the exceptional pushforward functor $j_!$ does not exist as a functor between standard categories of (quasi)coherent sheaves.

Historically, two distinct approaches have been proposed to resolve this by enlarging the category of sheaves:

1. Deligne's Approach: Extends the category to pro-coherent sheaves (specifically, pro-systems indexed by $\mathbb{N}^{op}$) to construct $j_!$ [Har66, Appendix].
2. Clausen-Scholze Approach: Extends the category to solid modules within the framework of condensed mathematics, providing a robust treatment of $j_!$ in a derived setting [CS19a, Lecture VIII].

The relationship between these two constructions has remained unclear. This paper aims to bridge these approaches by establishing a precise comparison between Deligne's pro-sheaf construction and the Clausen-Scholze solid module construction.

Methodology and Technical Framework
The author constructs a natural comparison functor $\Phi$ and analyzes its properties within the context of condensed mathematics and topological modules.

1. Functor Construction:
   The paper defines a functor $\Phi: \text{Pro}_{\mathbb{N}}\text{Mod}(A) \to \text{Solid}(A)$, where $A$ is a commutative ring. This functor maps a pro-system $\varprojlim_{\mathbb{N}} M_i$ to the solid module associated with the inverse limit of the underlying topological modules (equipped with the inverse limit topology).

   • The construction proceeds in two steps:
     • $\Phi_1$: Maps pro-systems to topological modules ($\text{TopMod}(A)_{cgwh}$).
     • $\Phi_2$: Maps these topological modules to condensed modules via the Yoneda embedding ($M \mapsto \text{Hom}_{\text{Top}}(-, M)$).
   • The composite $\Phi = \Phi_2 \circ \Phi_1$ factors through the category of solid modules, $\text{Solid}(A)$.

2. Analysis of Full Faithfulness:
   A central technical result involves the restriction of $\Phi$ to the subcategory of Mittag-Leffler pro-systems ($\text{Pro}^{ML}_{\mathbb{N}}\text{Mod}(A)$).

   • The author proves that $\Phi$ is fully faithful when restricted to Mittag-Leffler systems.
   • The proof relies on showing that for a Mittag-Leffler system, the canonical map from the colimit of homomorphisms to the homomorphisms of the limit is a bijection. This requires careful analysis of the inverse limit topology, specifically utilizing the fact that for countable index sets, the Mittag-Leffler condition ensures the projection maps stabilize onto the stable image.
   • The paper explicitly notes that this property fails for uncountable index sets or non-Mittag-Leffler systems, providing a counterexample involving $\omega_1$-indexed systems where the limit vanishes despite the system being Mittag-Leffler.

3. Derived Functor and Compatibility:
   The paper considers the right derived functor $R\Phi: D(\text{Pro}_{\mathbb{N}}\text{Mod}(A)) \to D(\text{Solid}(A))$. The main theorem establishes the compatibility of $R\Phi$ with the $j_!$ functors defined by Deligne ($j^{De}_!$) and Clausen-Scholze ($j^{CS}_!$).

Key Results
The primary result is Theorem 1.1 (detailed in Propositions 2.14 and 3.3):

• Existence and Properties of $\Phi$: There exists a natural functor $\Phi: \text{Pro}_{\mathbb{N}}\text{Mod}(A) \to \text{Solid}(A)$. This functor is left exact. Its restriction to the subcategory of Mittag-Leffler pro-systems, $\Phi_{ML}$, is exact and fully faithful.

• Compatibility of $j_!$: For a finite type $\mathbb{Z}$-algebra $A$ and a nonzerodivisor $f \in A$, let $j: \text{Spec}(A[1/f]) \hookrightarrow \text{Spec}(A)$ be the open immersion. The diagram involving the derived functors commutes:
  $$\begin{array}{ccc} \text{Mod}^{fp}(A[1/f]) & \xrightarrow{j^{De}_!} & D(\text{Solid}(A[1/f])) \\ \downarrow & & \downarrow \\ \text{Pro}_{\mathbb{N}}\text{Mod}^{fp}(A) & \xrightarrow{R\Phi} & D(\text{Solid}(A)) \end{array}$$
  Specifically, $R\Phi \circ j^{De}_! \cong j^{CS}_! \circ R\Phi$ for finitely presented modules.

• Proof Mechanism for Compatibility: The proof involves explicit computations of both sides of the commutative diagram.

  • Deligne's $j^{De}_!$ is realized via the cone of the map from a finitely presented module $M'$ to its $f$-adic completion.
  • Clausen-Scholze's $j^{CS}_!$ is computed using the solid tensor product and the resolution of $A[1/f]$ in the category of solid modules over $(A[T], A)$.
  • The author demonstrates that both constructions yield isomorphic objects in the derived category of solid modules by analyzing the $f$-adic completion $\widehat{M'}$ and its localization, utilizing flatness results for the map $A \to \widehat{A}$ and the localization $A \to A[1/f]$.

Significance and Claims
The paper claims to provide a "conceptual and structural unification" of the two approaches to the missing $j_!$ functor. By proving that Deligne's classical construction via pro-sheaves coincides with the modern Clausen-Scholze construction via solid modules, the work reinforces the compatibility of classical techniques with condensed mathematics tools.

The author notes that while the theorem is currently stated for a specific type of open immersion (defined by a nonzerodivisor in a finite type algebra), the result offers an "interesting idea" for the general case. The paper does not claim to have solved the general case for arbitrary open immersions but states an expectation that the theorem holds more generally, with the plan to "glue the functor $R\Phi$" in forthcoming work.

The work is modest in its scope, focusing on establishing the bridge between the two specific frameworks rather than proposing new experimental applications or extending the theory to entirely new geometric contexts beyond the comparison of these two formalisms.