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arxiv: 2606.10057 · v2 · pith:E6HDOQWMnew · submitted 2026-06-08 · 🧮 math.AG · math.NT

Ind-Banach approach to Grothendieck duality in Rigid-analytic geometry

Arun Soor This is my paper

Pith reviewed 2026-06-27 14:34 UTC · model grok-4.3

classification 🧮 math.AG math.NT
keywords Grothendieck dualityrigid-analytic geometryquasi-coherent sheavesInd-Banach spacesvolume formsquasi-compactly supported cohomologypartially-proper morphisms
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The pith

Duality theorem for quasi-compactly supported cohomology on rigid-analytic spaces identifies the dualizing object with volume forms.

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

The paper establishes a Grothendieck duality theorem for quasi-coherent sheaves on rigid-analytic spaces. For a smooth and Kiehl partially-proper morphism, the quasi-compactly supported cohomology satisfies a duality relation to the cohomology on the target space. The dualizing object is identified explicitly with the sheaf of volume forms. The argument relies on functional analysis carried out with Ind-Banach spaces rather than other frameworks. This supplies a concrete duality that applies directly to the rigid-analytic category.

Core claim

We prove a duality theorem for quasi-compactly supported cohomology of quasi-coherent sheaves on rigid-analytic spaces, with respect to a smooth and Kiehl partially-proper morphism. This includes an identification of the dualizing object with volume forms. The functional analysis underlying our theory does not use condensed mathematics, but rather Ind-Banach spaces.

What carries the argument

Ind-Banach spaces supply the functional analysis that produces the duality isomorphism and identifies the dualizing sheaf with the sheaf of top differentials.

If this is right

  • Cohomology groups with quasi-compact support on the source space are dual to ordinary cohomology groups on the target space.
  • The sheaf of top-degree differentials functions as the dualizing object for the morphism.
  • The result applies whenever the morphism is smooth and Kiehl partially-proper, covering many standard maps in rigid geometry.
  • Explicit duality pairings become available for computing one cohomology group from the other.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Ind-Banach technique may extend duality statements to other classes of analytic spaces where condensed methods have not yet been applied.
  • Explicit volume-form identifications could simplify residue calculations or trace maps on rigid curves and higher-dimensional spaces.
  • Comparison with algebraic Grothendieck duality might become feasible by base change to the algebraic closure or by specialization.

Load-bearing premise

The functional analysis with Ind-Banach spaces extends successfully to the rigid-analytic setting and supports the duality identification without additional obstructions.

What would settle it

A concrete counterexample consisting of a smooth Kiehl partially-proper morphism of rigid-analytic spaces together with a quasi-coherent sheaf whose quasi-compactly supported cohomology fails to satisfy the predicted duality would disprove the claim.

read the original abstract

We prove a duality theorem for quasi-compactly supported cohomology of quasi-coherent sheaves on rigid-analytic spaces, with respect to a smooth and Kiehl partially-proper morphism. This includes an identification of the dualizing object with volume forms. The functional analysis underlying our theory does not use condensed mathematics, but rather Ind-Banach spaces, following Ben-Bassat--Kelly--Kremnizer. Nevertheless, our overall strategy is inspired by that of Clausen--Scholze in the complex-analytic setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves a duality theorem for the quasi-compactly supported cohomology of quasi-coherent sheaves on rigid-analytic spaces, relative to a smooth and Kiehl partially-proper morphism. The dualizing object is identified with volume forms. The proof employs Ind-Banach spaces in the style of Ben-Bassat--Kelly--Kremnizer rather than condensed mathematics, while following the overall strategy of Clausen--Scholze from the complex-analytic case.

Significance. If the central identification holds, the result supplies a non-condensed route to Grothendieck duality in rigid geometry and makes the dualizing object geometrically explicit as volume forms. The approach may therefore be of interest to readers who prefer classical functional-analytic tools over condensed mathematics.

major comments (2)
  1. [Main theorem and the Ind-Banach setup (around the statement of the duality isomorphism)] The load-bearing step is the claim that the Ind-Banach completed tensor product and Hom functors commute with the relevant colimits on the rigid site so that the duality pairing for quasi-compactly supported cohomology is preserved. This is asserted in the main theorem but the verification that no new obstructions arise (in contrast to the condensed setting) must be exhibited explicitly; without it the identification of the dualizing object with volume forms does not follow.
  2. [Proof of the duality theorem] The manuscript must confirm that the Kiehl partially-proper hypothesis is used precisely to guarantee that the relevant pushforwards remain within the Ind-Banach category and that the quasi-compact support condition interacts correctly with the completed Hom; any gap here would restrict the stated generality.
minor comments (2)
  1. [Notation and preliminaries] Clarify the precise relationship between the rigid site and the Ind-Banach modules at the level of the site morphisms; the current notation for the completed tensor product could be made more uniform.
  2. [Introduction] Add a short comparison paragraph with the corresponding statement in Clausen--Scholze to highlight where the Ind-Banach argument diverges technically.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting the key aspects of our work. We are pleased that the significance is recognized. Below we address the major comments in detail, providing clarifications and indicating where revisions will strengthen the manuscript.

read point-by-point responses

  1. Referee: [Main theorem and the Ind-Banach setup (around the statement of the duality isomorphism)] The load-bearing step is the claim that the Ind-Banach completed tensor product and Hom functors commute with the relevant colimits on the rigid site so that the duality pairing for quasi-compactly supported cohomology is preserved. This is asserted in the main theorem but the verification that no new obstructions arise (in contrast to the condensed setting) must be exhibited explicitly; without it the identification of the dualizing object with volume forms does not follow.

    Authors: We acknowledge that an explicit verification of the commutation properties is essential for the rigor of the main result. In the manuscript, the commutation is used implicitly through the properties of Ind-Banach spaces established in Sections 3 and 4. However, to address the referee's point directly, we will add an explicit lemma (new Lemma 4.8) that verifies the commutation with colimits on the rigid site, showing that no new obstructions arise due to the specific nature of the Ind-Banach completion over non-archimedean fields. This will solidify the identification of the dualizing object with volume forms. revision: yes

  2. Referee: [Proof of the duality theorem] The manuscript must confirm that the Kiehl partially-proper hypothesis is used precisely to guarantee that the relevant pushforwards remain within the Ind-Banach category and that the quasi-compact support condition interacts correctly with the completed Hom; any gap here would restrict the stated generality.

    Authors: The Kiehl partially-proper condition is crucial in our setup, as explained in the introduction and used in the proof to keep pushforwards within the Ind-Banach category. We will revise the proof section to include a more detailed breakdown: specifically, we will add remarks after each application of the hypothesis in the proof of the main theorem to confirm how it ensures the pushforwards stay in the category and how the quasi-compact support interacts with the completed Hom via the established adjunctions. This will not restrict the generality but rather clarify it. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation builds on external Ind-Banach framework

full rationale

The paper states a duality theorem for quasi-compactly supported cohomology on rigid-analytic spaces using Ind-Banach spaces from Ben-Bassat--Kelly--Kremnizer (external citation) and draws inspiration from Clausen--Scholze without reducing the identification of the dualizing object with volume forms to any self-definition, fitted parameter, or self-citation chain. No equations or load-bearing steps in the provided abstract or description collapse the central claim to its inputs by construction; the approach is presented as an independent extension.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no information provided on free parameters, axioms, or invented entities used in the proof.

pith-pipeline@v0.9.1-grok · 5601 in / 998 out tokens · 18698 ms · 2026-06-27T14:34:12.366690+00:00 · methodology

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