Topological Vector Spaces

Technical Summary: Topological Vector Spaces in Condensed Mathematics

Problem and Motivation
The paper addresses the need to geometrize $p$-adic period morphisms for rigid analytic varieties, specifically to facilitate duality theorems for $p$-adic pro-étale cohomology. While Topological Vector Spaces (TVS) were previously introduced in [13] as enriched presheaves on "sympathetic" spaces valued in locally convex vector spaces over $\mathbb{Q}_p$, this paper investigates the condensed version of this category. The primary motivation, as stated in the introduction, is the application to duality results in [11], where the relevant examples of TVS arise from the cohomology of quasi-coherent sheaves on the Fargues-Fontaine curve, objects that naturally reside in the condensed universe.

The authors aim to establish a rigorous framework where:

1. Locally convex topological vector spaces are replaced by condensed (or solid) $\mathbb{Q}_p$-modules.
2. Sympathetic spaces are replaced by strictly totally disconnected spaces (or w-contractible spaces).
3. The enrichment condition (sensitivity to topology on mapping spaces) is formalized using the enriched Yoneda Lemma within the setting of condensed mathematics.

Methodology
The paper constructs and analyzes three nested categories of (pre)sheaves on the category of perfectoid affinoids over $C$ (where $C$ is the $p$-adic completion of an algebraic closure of a mixed-characteristic field $K$):

1. Vector Spaces (VS): The category of sheaves of $\mathbb{Q}_p$-vector spaces on the pro-étale site of perfectoid affinoids ($S_{\text{pro\'et}}$). These are algebraic objects without intrinsic topology, though the site encodes a topology.
2. Naive Topological Vector Spaces (NTVS): The category of sheaves of topological $\mathbb{Q}_p$-vector spaces (in the sense of Clausen-Scholze) on the same site. These are sheaves with values in condensed or solid abelian groups. The authors prove these form Grothendieck abelian categories and study their derived categories, solidification functors, and monoidal structures.
3. Topological Vector Spaces (TVS): The main object of study. This is the category of topologically enriched presheaves in NTVS.
   • The enrichment is achieved by equipping the Hom-spaces between perfectoid spaces with a natural condensed structure.
   • The authors restrict presheaf functors from general perfectoid affinoids to strictly totally disconnected spaces to ensure the enrichment condition holds and to utilize the enriched Yoneda Lemma.
   • A key technical tool is the distinction between "rigid" presheaves (where structure maps $F(Y \times T) \to [T, F(Y)]$ are isomorphisms) and general enriched presheaves. The paper establishes that sheaves arising from the pro-étale site are canonically rigid.

The methodology relies heavily on the theory of condensed mathematics (Clausen-Scholze), specifically the properties of solid modules, the relationship between condensed and solid categories, and the use of $\infty$-categories to handle derived functors and enriched structures.

Key Contributions and Results

1. Categorical Framework: The paper defines the category of TVS as a full subcategory of topologically enriched presheaves. It proves that this category, along with the categories of VS and NTVS, forms a Grothendieck abelian category with compact, projective generators (specifically, Yoneda objects tensored with solid modules).

2. Enriched Fully-Faithfulness (Theorem 1.1, Part 1):
   The authors prove that the canonical functor from Vector Spaces (VS) to Topological Vector Spaces (TVS) is fully faithful in the enriched sense, provided certain finiteness conditions are met.

   • Specifically, for $F \in D^b(S_{\text{pro\'et}}, \mathbb{Q}_p)$ and $G \in D^+(S_{\text{pro\'et}}, \mathbb{Q}_p)$, the canonical morphism
     $$R\pi_* R\text{Hom}_{S, \mathbb{Q}_p}(F, G) \to R\text{Hom}_{S_{\text{top}}, \mathbb{Q}_p}(R\pi_* F, R\pi_* G)$$
     is a quasi-isomorphism.
   • This result relies on the fact that algebraic Yoneda sheaves map to topological ones and that the enriched Yoneda Lemma holds in both settings.

3. Fargues-Fontaine Fully-Faithfulness (Theorem 1.1, Part 2):
   The paper establishes that the functor from quasi-coherent sheaves on the Fargues-Fontaine curve ($X_{FF}$) to TVS is fully faithful when restricted to perfect complexes.

   • For perfect complexes $F, G \in \text{Perf}(X_{FF, S^\flat})$, the natural morphism
     $$R\tau_* R\text{Hom}_{\text{QCoh}}(F, G) \to R\text{Hom}_{S_{\text{top}}, \mathbb{Q}_p}(R\tau_* F, R\tau_* G)$$
     is a quasi-isomorphism.
   • This is a topological analog of a result by Anschütz-Le Bras [3], reduced to the algebraic case via the first claim of the theorem.

4. Compatibility of Projections (Theorem 1.1, Part 3):
   The paper demonstrates the compatibility between the algebraic projection (from Fargues-Fontaine sheaves to VS) and the topological projection (to TVS) when restricted to nuclear sheaves. This ensures that the diagram relating the Fargues-Fontaine curve, VS, and TVS commutes for nuclear objects.

5. Computational Consequences (Corollary 1.2 & Example 4.27):
   The authors provide explicit computations of extension groups in the TVS category, showing they match those in VS and on the Fargues-Fontaine curve. For instance:

   • $R\text{Hom}_{S_{\text{top}}, \mathbb{Q}_p}(\mathbb{Q}_p, \mathbb{Q}_p) \simeq \mathbb{Q}_p$
   • $R\text{Hom}_{S_{\text{top}}, \mathbb{Q}_p}(\mathbb{Q}_p, \mathbb{G}_a) \simeq \mathbb{G}_a$
   • $R\text{Hom}_{S_{\text{top}}, \mathbb{Q}_p}(\mathbb{G}_a, \mathbb{G}_a) \simeq \mathbb{G}_a \oplus \mathbb{G}_a[-1]$
     These results confirm that extensions of Banach-Colmez spaces or sheaves on the Fargues-Fontaine curve can be computed interchangeably in the algebraic, pro-étale, or topological settings.

Significance and Claims
The paper claims that its primary significance lies in providing a robust condensed framework for Topological Vector Spaces. By replacing locally convex spaces with condensed/solid modules and sympathetic spaces with strictly totally disconnected spaces, the authors enable the use of powerful tools from condensed mathematics (such as the enriched Yoneda Lemma and solid tensor products) to study $p$-adic cohomology.

The authors explicitly state that this framework is motivated by and essential for the duality theorems in [11]. They note that the "solid version" of TVS allows for objectwise computations in the category of solid $\mathbb{Q}_p$-vector spaces, simplifying the analysis of cohomological dualities. The paper does not claim to solve the duality theorems themselves but rather provides the necessary categorical infrastructure (the "Topological Vector Spaces" category) to make such computations rigorous and tractable within the condensed universe. The results confirm that the "geometrization" of period morphisms via TVS preserves the essential homological information of the underlying algebraic and geometric objects.