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Compactly supported $p$-adic pro-\'etale cohomology of analytic varieties

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Pith's one-line read The paper establishes a boundary-fiber definition of compactly supported p-adic pro-étale cohomology for smooth partially proper rigid analytic varieties, and proves it agrees with the established compactly supported étale cohomology and…

desk verdict A coherent compact-support extension of the Colmez–Nizioł comparison package; the central theorems hold up, and the main caveats are explicitly flagged by the authors. read the letter →

arxiv 2501.13651 v1 pith:WDEVXG7E submitted 2025-01-23 math.AG math.NT

Piotr Achinger , Sally Gilles , Wies{l}awa Nizio{l} This is my paper · ORCID
classification math.AGmath.NT MSC 14F3014G2214F20
keywords p-adiccohomologypro-étaletopologyrigidanalyticvarietiessyntomicHyodo-KatocompactsupportSteinperiodmorphism
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

For smooth partially proper rigid analytic varieties over the p-adic completion $C$ of an algebraic closure, this paper establishes that compactly supported p-adic pro-étale cohomology should be defined as the homotopy fiber of restriction to the boundary pro-object $\partial X = \{X\setminus Z\}_{Z \text{ quasi-compact open}}$. With that definition, compactly supported pro-étale cohomology is canonically isomorphic, after inverting $\ell$, to the previously defined compactly supported étale cohomology of adic spaces; and for $r\ge 0$ the period morphism identifies $\tau_{\le r}$ of it with compactly supported syntomic cohomology built from Hyodo-Kato and $B^+_{\mathrm{dR}}$ cohomology. The paper also proves Poincaré duality for compactly supported de Rham and Hyodo-Kato cohomology, constructs compatible trace maps, and uses a conditional fundamental diagram to compute the groups for affine spaces, Stein curves, tori, and Drinfeld spaces. For a reader interested in duality or in explicit p-adic cohomology computations, this supplies the missing compact-support half of the comparison story.

What carries the argument

The load-bearing mechanism is the boundary-fiber construction: for any cohomology functor $F$ one forms $F_c(X) := [F(X) \to F(\partial X)]$ with $\partial X = \{X\setminus Z\}_{Z\in\Phi_X}$, turning compact support into a homotopy limit over quasi-compact complements. This makes excision and pushforward along relatively partially proper opens formal, and it is the bridge to the classical compact-support theory and to syntomic cohomology. In the derived $\infty$-category of solid $\mathbb{Q}_p$-modules, the syntomic side is expressed through the twisted Hyodo-Kato complex $[R\Gamma_{\mathrm{HK},c}(X)\otimes^{\mathbb{L}}_{F^{\mathrm{nr}}} B^+_{\mathrm{st}}]_{N=0,\phi=p^r}$ and the quotient $R\Gamma_{\mathrm{dR},c}(X/B^+_{\mathrm{dR}})/F^r$. Because truncation $\tau_{\le r}$ commutes with the colimits defining $\partial X$, the known comparison for $X$ and each $X\setminus Z$ upgrades to a comparison for supports. For Stein varieties the explicit computations additionally rely on a fundamental diagram whose validity is governed by surjectivity of maps $\pi^{r-1,r}_d$ and $\pi^{r-1,r-d}_1$, equivalently by Frobenius slope bounds on compactly supported Hyodo-Kato cohomology.

What would settle it

Compute the Frobenius slopes on $H^{r-1}_{HK,c}(X_C)$ for a smooth Stein variety $X$ of dimension $d$ and compare them with the interval $\{r-d-1,\, r-d\}$; any slope outside this interval makes $\pi^{r-1,r}_d$ non-surjective by Remark 8.6, so the exact sequence of Theorem 8.4 would not be available for that $X$.

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Extended reading notes

Core claim

The central claim is that the object $R\Gamma_{\mathrm{pro\acute{e}t},c}(X,\mathbb{Q}_p) := [R\Gamma_{\mathrm{pro\acute{e}t}}(X,\mathbb{Q}_p) \to \mathrm{colim}_{Z\in\Phi_X} R\Gamma_{\mathrm{pro\acute{e}t}}(X\setminus Z,\mathbb{Q}_p)]$ is a good compact-support theory: for $X$ partially proper it is rationally quasi-isomorphic to the established compactly supported étale cohomology of adic spaces, and the period morphism $\alpha_{r,c}$ from compactly supported syntomic cohomology is a quasi-isomorphism after $\tau_{\le r}$. The syntomic side is computed by the compactly supported version of the triangle $R\Gamma_{\mathrm{syn},c}(X,\mathbb{Q}_p(r)) \to [R\Gamma_{\mathrm{HK},c}(X)\otimes^{\mathbb{L}}_{F^{\mathrm{nr}}} B^+_{\mathrm{st}}]_{N=0,\phi=p^r} \to R\Gamma_{\mathrm{dR},c}(X/B^+_{\mathrm{dR}})/F^r$. On this foundation the paper builds trace maps compatible with the classical trace, dualities for de Rham and Hyodo-Kato cohomology, and explicit Stein-space computations; the overconvergent dagger analogues are shown to agree with the rigid analytic ones for partially proper varieties.

Load-bearing premise

For Stein varieties, the displayed fundamental diagram and the computations built on it require surjectivity of two maps from Frobenius-twisted compactly supported Hyodo-Kato cohomology to compactly supported de Rham cohomology, a condition the paper verifies for tori and Drinfeld spaces but not for all Stein varieties.

Editorial extensions

If this is right

  • For every smooth partially proper rigid analytic variety over $C$, compactly supported p-adic pro-étale cohomology is canonically identified, rationally, with the established compactly supported étale cohomology of adic spaces; finiteness and duality results for the latter therefore become available.
  • In degrees $r\ge 0$, the $\tau_{\le r}$ truncation of compactly supported pro-étale cohomology is computed by compactly supported syntomic cohomology, hence by Hyodo-Kato and $B^+_{\mathrm{dR}}$ data.
  • The trace map $H^{2d}_{\mathrm{pro\acute{e}t},c}(X,\mathbb{Q}_p(d))\to\mathbb{Q}_p$ exists on compactly supported pro-étale cohomology and agrees with both the syntomic trace and the classical étale trace; this is the pairing needed for Poincaré duality.
  • For Stein varieties satisfying the slope condition, the fundamental diagram expresses $H^r_{\mathrm{pro\acute{e}t},c}(X_C,\mathbb{Q}_p(r))$ in terms of compactly supported holomorphic forms and Frobenius-invariant pieces of compactly supported Hyodo-Kato cohomology.
  • Explicit computations for affine space, Stein curves, tori, and Drinfeld spaces give concrete compactly supported groups, with the representation-theoretic factors that are expected for Drinfeld spaces appearing in the results.

Reading between the lines

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

  • The boundary-fiber recipe is independent of $p$ and of the particular cohomology theory, so the same construction should yield a canonical compact-support formalism for other sheaves on taut rigid spaces, with the paper's comparison results as evidence.
  • If the Section 8 slope condition holds for general p-adic period domains—the paper suggests this is likely—the fundamental diagram would provide compactly supported pro-étale cohomology for period domains, connecting the computations to representation-theoretic applications.
  • The comparison is only asserted after $\tau_{\le r}$, and the affine-line example shows the failure is concentrated at the boundary; a testable expectation is that the boundary contribution to non-compact cohomology is exactly what the homotopy-fiber definition removes.
  • The existence of a lift of the period morphism to topological vector spaces suggests the comparison is compatible with base change along perfectoid spaces, opening the way to a relative or family version of the theorem.
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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

2 major / 4 minor

Summary. This paper develops a theory of compactly supported p-adic pro-étale cohomology for smooth partially proper rigid analytic varieties. It defines the compactly supported theory as the homotopy fiber of the map from RΓ_proét(X) to the boundary colimit colim_Z RΓ_proét(X\Z), proves that this agrees with Huber's compactly supported étale cohomology, and constructs compactly supported versions of de Rham, B_dR^+, Hyodo-Kato, and syntomic cohomologies. The main comparison theorem (Theorem 6.13) identifies τ≤r RΓ_syn,c(X,Q_p(r)) with τ≤r RΓ_proét,c(X,Q_p(r)) for all smooth X and r≥0. A conditional fundamental diagram for Stein varieties is established under explicit Frobenius-slope surjectivity conditions (Theorem 8.4), with applications to affine spaces, tori, Drinfeld spaces, and Stein curves.

Significance. If the results hold as stated, they provide a workable compactly supported p-adic pro-étale cohomology theory with the expected stable-range comparison isomorphism. The central comparison is obtained by reducing to the established Colmez–Nizioł comparison for usual cohomology and exploiting formal properties of boundary colimits; the paper contains no fitted parameters or tuned predictions. The explicit, honest treatment of the slope conditions in Section 8 is a strength, as it delineates exactly where the fundamental diagram is conditional. The paper should be of interest to researchers in p-adic Hodge theory and rigid analytic geometry.

major comments (2)
  1. [Section 2.2.3, Lemma 2.31] The comparison with Huber's rational compactly supported cohomology (Theorem 2.23) is a foundational claim, and its proof relies on Lemma 2.27, Lemma 2.28, and Corollary 2.29. The proofs of these topological statements are very compressed: for example, Corollary 2.29 asserts without a detailed argument that the constructed family of opens is cofiltering and cofinal with the family of quasi-compact opens. Since a gap here would affect the identification with Huber's cohomology, please expand these arguments or give precise references for each assertion.
  2. [Section 8.1, Theorem 8.4] The fundamental diagram is stated under the surjectivity of the maps π^{r-1,r}_d and π^{r-1,r-d}_1, which, as Remark 8.6 explains, is equivalent to a Frobenius-slope condition on compactly supported Hyodo-Kato cohomology. The paper verifies this condition only for tori and Drinfeld spaces. The conditional nature is acknowledged, but because the introduction and abstract present the fundamental diagram as a main result, please state the slope condition prominently in the abstract and in Corollary 1.6, and summarize the class of varieties for which the condition is known to hold.
minor comments (4)
  1. [Lemma 2.27] The statement 'Moreover, U is equal to the interior of U' appears to contain a typo or a confusing abuse of notation; presumably the intended statement is about the closure of U and the interior of its closure. Please rewrite the statement and clarify the definition of U' in the proof.
  2. [Proposition 4.9(1)(a)] The target of the morphism ϑ_c is written as RΓ_dR(X) in the displayed statement, but it should be RΓ_dR,c(X) to match the compactly supported setting and the analogous statement in Proposition 3.10(1).
  3. [Section 2.2.2] The phrase 'nous écrivons simplement Γ_c(X,F) pour Γ_c(...)' mixes French and English; replace 'pour' with 'for' or rewrite the sentence in English.
  4. [Theorem 6.13] The proof uses the fact that truncation τ≤r commutes with filtered colimits in D(Q_p,□). This is a standard formal property, but since it is load-bearing for the boundary terms, please add a precise statement or reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central comparison theorems are built on published external anchors, and the conditional slope assumptions are explicit limitations, not hidden inputs.

full rationale

I found no circular step in the paper's derivation chain. The main comparison theorem (Theorem 6.13) is proved by a standard distinguished-triangle argument: the map for compact support is induced functorially from the non-compact period morphism, and the boundary term is a filtered colimit of the same morphism. The only non-formal input is the cited comparison theorem [15, Cor. 7.3] for smooth rigid analytic varieties without compact support; that is a published, independently checkable result of the same authors, and the compact-support statement does not reduce to it by definition. Similarly, Theorem 2.23 is proved through the P(X) construction and Lemma 2.31, not by assuming the Huber comparison; the topological lemmas used there are stated with references and are not circular. The conditional slope-surjectivity conditions in Section 8.1 (Lemma 8.1, Proposition 8.3, Theorem 8.4) are explicitly presented as assumptions, with Remark 8.6 noting when they hold, and they are verified for tori and Drinfeld spaces; this is a limitation of the Stein computations, not a circularity. The paper's repeated reliance on [15], [16], and [10] is real self-citation, but those citations are load-bearing, published results with stated assumptions that do not include the target compact-support statements, so under the review rules they count as independent support and do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters were fitted and no new physical or mathematical entities were postulated. The new objects, such as compactly supported pro-étale cohomology, are definitions built from existing cohomology theories. The main external inputs are prior results, some from unpublished preprints.

assumptions (6)
  • domain assumption Condensed mathematics and solid modules framework (Scholze).
    Used throughout; the paper works in the derived ∞-category of solid Qp-modules and assumes properties such as solid tensor products commuting with colimits and flatness of B+_st. See Notation and Sections 2 to 6.
  • domain assumption Colmez-Nizioł comparison theorems for non-compactly-supported cohomology [15,16].
    Black boxes: the period morphism RΓ_syn(X,Qp(r)) to RΓ_proét(X,Qp(r)) is a quasi-isomorphism after τ≤r, and the Hyodo-Kato isomorphisms hold in the condensed setting. Invoked throughout, especially in Theorems 6.13 and 6.14.
  • domain assumption Huber's compactly supported étale cohomology and its properties [22,23].
    Provides the benchmark definition, torsion finiteness, trace maps, and spectral sequences used in Section 2 and Section 7.
  • domain assumption Serre and coherent duality for rigid analytic and dagger varieties (Chiarellotto [8], van der Put [36], Grosse-Klönne [21]).
    Basis for the de Rham and Hyodo-Kato duality results in Section 5.
  • domain assumption Fujiwara-Kato foundations of rigid geometry [19].
    Used for topological properties of closures and interiors in partially proper spaces in Lemmas 2.27 and 2.28 and Corollary 2.29.
  • domain assumption GAGA-type comparison theorems from the preprint [34] and trace compatibility from [40].
    Used only in Proposition 7.17 to identify the pro-étale trace map with Huber's trace map by reduction to the algebraic case; not reproduced in this paper.

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Pith. "Pith review of Compactly supported $p$-adic pro-\'etale cohomology of analytic varieties." pith.science (2026). https://pith.science/paper/WDEVXG7E

@misc{pith2026250113651,
  author       = {Pith},
  title        = {Pith review of: Compactly supported $p$-adic pro-\'etale cohomology of analytic varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDEVXG7E}},
  note         = {Machine review of arXiv:2501.13651}
}
abstract

We study properties of compactly supported $p$-adic pro-\'etale cohomology of smooth partially proper rigid analytic varieties. In particular, we prove a comparison theorem, in a stable range, with compactly supported syntomic cohomology, which is built from compactly supported Hyodo-Kato and ${\mathcal B}^+_{\rm dr}$-cohomologies. We derive from that a (limited version of a) fundamental diagram.

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Cited by 1 Pith paper

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    math.AG 2025-09 conditional novelty 6.0 of 10

    A condensed version of the Topological Vector Spaces category is introduced, and fully faithful embeddings from algebraic p-adic vector spaces and from perfect complexes on the Fargues–Fontaine curve are proven.

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