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Topological Vector Spaces

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Two p-adic sheaf categories fit inside Topological Vector Spaces

desk verdict A real new framework with two genuinely new fully faithfulness theorems; the final dévissage in the Fargues–Fontaine result has a fixable proof gap that a referee should flag. read the letter →

arxiv 2509.25981 v3 pith:ALJ3FUWC submitted 2025-09-30 math.AG math.FAmath.NT

Pierre Colmez , Wies{l}awa Nizio{l} This is my paper · ORCID
classification math.AGmath.FAmath.NT MSC 14G4514F3018F20
keywords TopologicalVectorSpacescondensedmathematicsFargues–Fontainecurveperfectoidpro-étalecohomologyfullyfaithfulembeddingsBanach–Colmezp-adicHodgetheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that a single category—Topological Vector Spaces, built from condensed Q_p-modules with a continuity condition on restriction maps—contains two previously separate worlds as full subcategories: the algebraic Vector Spaces used for p-adic pro-étale cohomology, and perfect complexes on the relative Fargues–Fontaine curve. If true, extension groups and Hom-computations can be carried out in one setting and transferred freely to the other, which is what duality theorems for p-adic cohomology of rigid analytic spaces need. The main theorem states three things: the functor from Vector Spaces to Topological Vector Spaces is enriched fully faithful under mild finiteness hypotheses; the functor from perfect complexes on the Fargues–Fontaine curve to Topological Vector Spaces is fully faithful; and the algebraic and topological projections agree on nuclear sheaves. A sympathetic reader would take away that the topological category is not a new object with new answers, but a home where existing computations become flexible.

What carries the argument

The load-bearing device is topological enrichment: presheaves are required to have restriction maps that are continuous for the natural condensed structure on mapping spaces between perfectoid affinoids, and the Hom between enriched presheaves is defined as an enriched end. The enriched Yoneda Lemma then lets the authors treat these enriched presheaves homologically exactly as if they were ordinary algebraic presheaves. For the Fargues–Fontaine part, the proof reduces every vector bundle, after étale localization and twist, to the two basic sheaves O and the skyscraper at infinity i_{∞,*}R_S, and uses the identity Rτ'_*(i_{∞,*}R_S)≃G_a to transfer the known algebraic fully faithfulness to th

What would settle it

Compute RΓ(X_{FF,Y^♭}, i_{∞,*}R_S) for a strictly totally disconnected perfectoid Y over S; the claimed reduction requires it to equal R_Y = G_a(Y) in degree 0 with no higher cohomology. Any deviation from that quasi-isomorphism would break the bridge from the algebraic Fargues–Fontaine pushforward to Corollary 4.7.

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Extended reading notes

Core claim

The central claim is Theorem 1.1. For S a strictly totally disconnected perfectoid affinoid over C, the canonical functor Rπ_* from p-adic sheaves on the pro-étale site of S (Vector Spaces) to Topological Vector Spaces is enriched fully faithful: whenever F is bounded and Rπ_*F is bounded, and G is bounded below, the natural map Rπ_*R Hom_S(F,G) → R Hom_{S^top}(Rπ_*F,Rπ_*G) is a quasi-isomorphism. Likewise, the functor Rτ_* from quasi-coherent sheaves on the relative Fargues–Fontaine curve to Topological Vector Spaces is fully faithful when restricted to perfect complexes, meaning internal Homs between perfect complexes are computed identically in the two categories. Finally, on nuclear shea

Load-bearing premise

The proof of fully faithfulness for perfect complexes reduces every vector bundle, after étale localization and twists, to the two sheaves O and the skyscraper at infinity, and then asserts without proof that the algebraic pushforward of that skyscraper is the additive group G_a; if that identity is false, the reduction to the Q_p/G_a case does not go through.

Editorial extensions

If this is right

  • Ext groups between Banach–Colmez spaces or between perfect complexes on the Fargues–Fontaine curve can be computed in Topological Vector Spaces with the same answer as in the algebraic categories.
  • The enriched fully faithfulness applies to complexes representing p-adic pro-étale cohomology of smooth partially proper rigid analytic varieties, so duality statements can be formulated in the topological category.
  • The compatibility on nuclear sheaves means computations involving nuclear sheaves can be made in whichever of the algebraic or topological projections is more convenient.
  • Explicit Ext computations: RHom(Q_p,Q_p)=Q_p, RHom(Q_p,G_a)=G_a, RHom(G_a,G_a)=G_a⊕G_a[-1], RHom(G_a,Q_p)=G_a(-1)[-1], all valid in the topological category.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The unproved assertion inside Corollary 4.25 that Rτ'_*(i_{∞,*}R_S) equals G_a is the bridge linking the Fargues–Fontaine pushforward to the Q_p/G_a case; a reader should treat this as the point to check, since the rest of the perfect-complex proof leans on it.
  • A natural test is whether fully faithfulness extends to pseudocoherent complexes or to all nuclear sheaves; the authors themselves note the statement is more general than perfect complexes, without a clean formulation.
  • The same enrichment mechanism could plausibly be applied with other coefficients (integral p-adic sheaves, finite coefficients) or on the étale site, where the analogous fully faithfulness statements are not explored here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a condensed-mathematics framework for Topological Vector Spaces (TVS) and Naive Topological Vector Spaces (NTVS) over perfectoid spaces, and proves three main results: (1) an enriched fully faithfulness of the canonical functor Rπ_* from the pro-étale derived category of a strictly totally disconnected perfectoid space S to the derived category of topological sheaves, under certain finiteness hypotheses; (2) full faithfulness of the functor Rτ_* from perfect complexes on the relative Fargues-Fontaine curve X_{FF,S^♭} to TVS; and (3) a compatibility of the algebraic and topological projections on nuclear sheaves. The paper also establishes substantial foundational material: topological pro-étale sites, condensed versus solid sheaves, topologically enriched presheaves, an enriched Yoneda lemma, monoidal structures, and derived categories. The main theorems are intended to enable Ext computations in p-adic cohomology and duality theorems.

Significance. If the results hold, this is a significant contribution to p-adic geometry and condensed mathematics. The paper introduces a workable category of topologically enriched sheaves, proves that both algebraic Vector Spaces and perfect complexes on the Fargues-Fontaine curve embed fully faithfully into it, and supplies tools for computing extensions of Banach-Colmez spaces and vector bundles. The reduction of the Fargues-Fontaine fully faithfulness to the Anschütz–Le Bras theorem is conceptually clear and, if the missing details are supplied, would give a powerful bridge between algebraic and topological sheaf theory. The paper is also valuable for its detailed treatment of enrichment, solidification, and derived structures on presheaf categories.

major comments (3)
  1. [Section 4.2.4, proof of Corollary 4.25] The proof asserts in one sentence that Rτ'_*(i_{∞,*}R_S)=G_a ('the second quasi-isomorphism is clear'). This is load-bearing: it is what reduces the final statement to Corollary 4.7, which only treats the sheaves Q_p and G_a. The value of Rτ'_*(i_{∞,*}R_S) at Y involves cohomology of the structure sheaf on the tilt Y^♭, and identifying it with G_a is a nontrivial period-ring computation. Please provide a proof or a precise reference. If the identity requires additional hypotheses beyond S∈sPerf_C, those hypotheses must be stated and verified in the present setting.
  2. [Section 4.2.4, Eq. (4.27)] Even granting Rτ'_*(O)≃Q_p and Rτ'_*(i_{∞,*}R_S)=G_a, the invocation of Corollary 4.7 requires an identification Rτ_*E ≃ Rπ_*Rτ'_*E for the generators E∈{O, i_{∞,*}R_S}. The only result of this type is Lemma 4.21, which applies to nuclear sheaves. Neither O nor i_{∞,*}R_S is shown to be nuclear in the proof, and i_{∞,*}R_S, being supported on the boundary divisor, is not obviously nuclear. Without this identification, the right-hand side of (4.27) is not the topological Hom of Q_p and G_a, and Corollary 4.7 cannot be applied. Please prove the identification directly for these generators, or modify the argument so that (4.27) is reduced to Corollary 4.7 without this step.
  3. [Section 4.1.1, proof of Theorem 4.1] The reduction from a general G∈D^+(S_proét,Ab) to a single injective sheaf is asserted via 'a limit argument and a dévissage' with no details. This is a central step in the proof of the enriched fully faithfulness: the morphism (4.2) involves Rπ_* and RHom, which do not in general commute with the limits involved in reconstructing a bounded-below complex from its terms. The argument should be spelled out, for example using Postnikov towers and the finite cohomological amplitude hypothesis on Rπ_*F. As written, this leaves a gap in the proof of Theorem 4.1(1).
minor comments (5)
  1. [Throughout] There are several typographical errors: 'Anreychev' in the Introduction (footnote 2) should be 'Andreychev'; 'Anchütz' in reference [3] should be 'Anschütz'; 'cordinate' in the proof of Lemma 4.11 should be 'coordinate'.
  2. [Section 2.1.2] The diagram defining the maps of sites π and η is garbled in the text; please redraw it so that the directions of the arrows are clear.
  3. [Section 4.2.4] The notation R_S in i_{∞,*}R_S is never defined. Since this is a key object (the skyscraper sheaf on the boundary divisor of the Fargues-Fontaine curve), please define it precisely, presumably as the structure sheaf on S^♭ under the inclusion i∞: S^♭↪X_{FF,S^♭}.
  4. [Section 4.2.4, proof of Corollary 4.25] The reduction from line bundles to the set {O,O(1)} and then to {O,i_{∞,*}R_S} is very terse. In particular, the use of the Euler sequence and the exact sequence 0→O→O(1)→i_{∞,*}R_S→0 to dévissage RHom is only indicated. A brief explanation of why the desired statement is stable under extensions (e.g., via the five lemma) would improve readability.
  5. [References] Reference [30] is a MathOverflow post; if possible, replace it with a standard published reference for prodiscrete spaces, or give a more formal citation.

Circularity Check

0 steps flagged · score 1.0 of 10

No meaningful circularity: the central fully-faithfulness proofs are self-contained or reduce to independent published theorems; the only flagged issue is an omitted 'clear' computation in Corollary 4.25, which is a completeness gap rather than a circular reduction.

full rationale

Walking the derivation chain, I find no step in which a claimed output is identical by construction to an input. Theorem 4.1 is proven internally: π_* is factored through presheaves and the key comparison (4.6) is obtained from the classical and enriched Yoneda lemmas, with no reliance on the authors' prior work. Corollary 4.25 reduces algebraic fully faithfulness to Anschütz–Le Bras [3, Cor. 3.10] and then invokes Corollary 4.7; the local identifications Rτ'_*(O)≃Q_p and Rτ'_*(i_{∞,*}R_S)=G_a are cited to Fargues–Scholze or asserted ('Indeed, the second quasi-isomorphism is clear'). This is an unproved/under-justified computation, and the proof does not explicitly justify Rτ_*≃Rπ_*Rτ'_* on these generators outside the nuclear setting of Lemma 4.21; that is a real completeness/correctness gap, but it is not a circular reduction—the target quasi-isomorphism is not being used as its own input. Self-citations ([10],[11],[12],[1]) appear in the introduction, remarks, and an application (Corollary 4.8), not as inputs to the central fully-faithfulness proofs. The paper contains no fitted parameters, no predictions from fitted data, and no author-imported uniqueness theorem. Score 1 reflects only the presence of minor, non-load-bearing self-citations; the central mathematical content is independent.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central results rest on standard condensed mathematics and on published theorems of Anschütz–Le Bras, Fargues–Scholze, Kedlaya–Liu, and Scholze. No free parameters or fitted quantities appear. The only non-standard support is a MathOverflow citation for prodiscreteness.

assumptions (7)
  • domain assumption Condensed mathematics framework of Clausen–Scholze, including solidification functors and Proposition 2.7.
    The entire paper operates in condensed/solid abelian groups; these properties are imported from [8] without reproof.
  • domain assumption Anschütz–Le Bras algebraic fully faithfulness [3, Cor. 3.10]: Rτ'_* is fully faithful on perfect complexes.
    Used directly in the proof of Corollary 4.25 to identify RHom after algebraic pushforward.
  • domain assumption Fargues–Scholze structure theory of the Fargues–Fontaine curve: classification of vector bundles and properties of O(1), [13, Prop. II.3.1, Cor. II.2.20, Prop. II.2.5].
    Used in Corollary 4.25 to reduce perfect complexes to twists of O and the skyscraper at infinity.
  • domain assumption Kedlaya–Liu [19, Prop. 8.4.7]: pro-étale Q_p-local systems admit Z_p-lattices after localization.
    Used to trivialize local systems appearing in the reduction to line bundles.
  • domain assumption Pro-étale acyclicity of strictly totally disconnected perfectoid spaces (Scholze [26]) and Tate acyclicity for G_a.
    Used in Lemma 2.5 and Remark 4.9 to ensure π_*-acyclicity of Q_p and G_a.
  • standard math Existence of K-injective and K-flat resolutions in Grothendieck abelian categories [24], [29].
    Foundational for derived internal Hom and tensor products used throughout.
  • standard math Point-set fact that Hom_{O_C}(R_2^+,R_1^+) is prodiscrete, cited to a MathOverflow answer [30].
    Needed in Lemma 3.2 for prodiscreteness of mapping spaces; the source is non-traditional.

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Pith. "Pith review of Topological Vector Spaces." pith.science (2026). https://pith.science/paper/ALJ3FUWC

@misc{pith2026250925981,
  author       = {Pith},
  title        = {Pith review of: Topological Vector Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ALJ3FUWC}},
  note         = {Machine review of arXiv:2509.25981}
}
abstract

Motivated by applications to duality theorems for $p$-adic pro-\'etale cohomology of rigid analytic spaces, we study the category of Topological Vector Spaces in the setting of condensed mathematics. We prove that it contains, as full subcategories, both the category of (topologically) bounded algebraic Vector Spaces and the category of perfect complexes on the Fargues-Fontaine curve. Vector Spaces coming from $p$-adic pro-\'etale cohomology of smooth partially proper rigid analytic varieties are examples of sheaves belonging to the former category.

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