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An integral comparison of crystalline and de Rham cohomology

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Pith's one-line read The paper proves an integral, untwisted comparison between twisted crystalline and de Rham cohomology for smooth proper formal $\mathcal{O}_K$-schemes, with coefficients in perfect complexes of prismatic $F$-crystals.

desk verdict The paper proves a genuinely new integral Berthelot–Ogus comparison with coefficients for arbitrary ramification, but a load-bearing 'crystal property' is asserted rather than proved. read the letter →

arxiv 2507.17631 v1 pith:YACTXLL5 submitted 2025-07-23 math.NT math.AG

Abhinandan , Alex Youcis This is my paper · ORCID
classification math.NTmath.AG MSC 14F3014F40
keywords prismaticcohomologycrystallinedeRhamBerthelot–Oguscomparisonintegralp-adicHodgetheoryF-crystalstorsioninBreuil–Kisinmodules
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

Berthelot and Ogus compared de Rham and crystalline cohomology only after inverting $p$. This paper proves a version that holds integrally, without inverting $p$, provided both sides are twisted by enough powers of Frobenius: $n \geq a = \lceil \log_p(e/(p-1))\rceil$, where $e$ is the ramification index of $K/\mathbb{Q}_p$. The comparison is with coefficients in any perfect complex of prismatic $F$-crystals on a smooth proper formal $\mathcal{O}_K$-scheme. The proof uses a prismatic analogue of Dwork's trick: the Frobenius on the prismatization of $\mathcal{O}_K$ lets one shrink the 'disk' $\mathrm{Spf}(\mathcal{O}_K)$ until the twisted de Rham and crystalline points coincide. A rational untwisting then recovers the classical Berthelot\--Ogus isomorphism with coefficients.

What carries the argument

The stacky prismatization $\mathcal{O}^\Delta_K$ (the formal stack whose structure sheaf carries a Frobenius lift), together with the two maps $\rho^{(n)}_{\mathrm{dR}}$ and $\rho^{(n)}_{\mathrm{crys}} : \mathrm{Spf}(\mathcal{O}_K), \mathrm{Spf}(W) \to \mathcal{O}^\Delta_K$ obtained by precomposing with $F^n$. The prismatic Dwork trick is Theorem 3.12 and Proposition 3.25: for $n \geq a$ the composition $\mathrm{Spf}(\mathcal{O}_K) \to \mathrm{Spf}(W) \xrightarrow{\rho^{(n)}_{\mathrm{crys}}} \mathcal{O}^\Delta_K$ is identified with $\rho^{(n)}_{\mathrm{dR}}$. This identification is realised through the modified Breuil prism $(\widetilde{S}, p)$, where $\widetilde{S} = S\{\varphi(u^{\widetilde{e}})/p\}^\wedge_\delta$, and a diagram (5) whose lower arrows express the constancy of the pullback after restriction to the smaller subdisk.

What would settle it

Compute, for a smooth proper formal scheme $X/\mathcal{O}_K$ and a perfect prismatic $F$-crystal $V$ that is not a vector bundle, the $p$-adic torsion lengths of $H^i_{(n),\mathrm{dR}}(X,V)$ and $H^i_{(n),\mathrm{crys}}(X_k,V)$ for some $n \geq a$; if $\ell(H^i_{(n),\mathrm{dR}}[p^\infty]) \neq e\cdot \ell(H^i_{(n),\mathrm{crys}}[p^\infty])$ for any $(i,n)$, Theorem 4.9 fails. Equivalently, find a perfect complex for which the crystal property at the mixed Breuil–Kisin prism fails, which would break the independence of the uniformiser in Construction 3.19.

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Extended reading notes

Core claim

The paper's central claim is Theorem 4.9: for $V$ a perfect prismatic crystal on $X$, there is a natural integral generalised Berthelot\--Ogus isomorphism $R\Gamma^{(n)}_{\mathrm{crys}}(V)\otimes_W \mathcal{O}_K \simeq R\Gamma^{(n)}_{\mathrm{dR}}(V)$ for every $n \geq a = \lceil \log_p(e/(p-1))\rceil$. Corollary 4.16 untwists the Frobenius rotations rationally and recovers an isomorphism $R\Gamma_{\mathrm{crys}}(V_{\mathrm{crys}})\otimes_W K \simeq R\Gamma_{\mathrm{dR}}(V_{\mathrm{dR}})\otimes_{\mathcal{O}_K} K$, extending Berthelot\--Ogus to coefficients in perfect prismatic $F$-crystals. The authors view the proof as a prismatic incarnation of Dwork's trick: via the stacky prismatization, the missing Frobenius on $\mathrm{Spf}(\mathcal{O}_K)$ is replaced by the Frobenius of the stack $\mathcal{O}^\Delta_K$, and for $n \geq a$ the $n$-twisted de Rham point and the $n$-twisted crystalline point of $\mathcal{O}^\Delta_K$ become identified (Theorem 3.12, refined by Proposition 3.25), forcing the cohomological comparison.

Load-bearing premise

The proof assumes that perfect complexes of prismatic crystals satisfy the crystal property—that pullback along the two projections of the mixed Breuil–Kisin prism $S_{\pi,\pi'}$ yields an identification of perfect complexes—in the derived sense needed for the independence of the uniformiser; this is invoked without proof in Construction 3.19 and diagram (3.11) of Proposition 3.23.

Editorial extensions

If this is right

  • For $n \geq a$ the integral comparison gives a $W$-descent for $n$-twisted de Rham cohomology: it depends only on the special fibre $X_k$ and the restriction of the crystal, not on the full formal scheme.
  • The equality $\ell^{(n)}_{\mathrm{dR}} = e\cdot \ell^{(n)}_{\mathrm{crys}}$ for $n \geq a$ (combining Proposition 4.24 and (4.4)) turns the study of torsion in the two classical cohomologies into the study of $u^\infty$-torsion in Breuil\--Kisin cohomology.
  • Conjecture 4.19 ($\ell_{\mathrm{crys}} \leq \ell_{\mathrm{dR}} \leq e\cdot \ell_{\mathrm{crys}}$) is reduced, under Hypothesis 4.28, to finiteness and monotonicity of $u^\infty$-torsion, and is verified in height $i \leq 2$ cases in Appendix A.
  • The rational untwisting (Corollary 4.16) yields a coefficient version of Berthelot\--Ogus valid for all ramification degrees $e$, not just $e \leq p-1$.

Reading between the lines

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

  • The threshold $a = \lceil \log_p(e/(p-1))\rceil$ behaves like a convergence radius: the paper proves the identification is sharp (the 'if and only if' in Proposition 2.13), so one may test numerically whether the integral comparison genuinely fails below $a$.
  • The same stacky Dwork-trick mechanism should transfer to other settings with a Frobenius-bearing stack—such as $q$-de Rham prisms or log-prismatic cohomology—once the analogous 'constancy' diagram is established.
  • The torsion framework suggests concrete experiments: for abelian schemes or complete intersections over wildly ramified fields, compute $\ell^i_{\mathrm{dR}}$ and $\ell^i_{\mathrm{crys}}$ in low degrees to test Conjecture 4.19; the Li\--Petrov example in the paper already shows strict inequality can occur.
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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 / 6 minor

Summary. The paper proposes an integral, coefficiented analogue of the Berthelot–Ogus comparison between crystalline and de Rham cohomology for a smooth proper formal scheme X over a mixed characteristic DVR O_K. Working in the stacky prismatic formalism of Drinfeld and Bhatt–Lurie, the authors define n-twisted crystalline and de Rham realisation functors and prove (Theorem 4.9) that for a perfect prismatic crystal V and n at least a = ceil(log_p(e/(p-1))), there is a natural isomorphism RΓ^(n)_crys(V) ⊗_W O_K ≃ RΓ^(n)_dR(V). They then rationally untwist this isomorphism (Corollary 4.16) to recover the classical Berthelot–Ogus isomorphism with coefficients in perfect complexes of prismatic F-crystals. The proof proceeds through a stack-theoretic comparison (Theorem 3.12) and a more explicit Breuil–Kisin-theoretic comparison (Propositions 3.23 and 3.25), with uniformiser-independence handled via a mixed Breuil–Kisin prism. The final section proposes a conjectural framework relating torsion in crystalline and de Rham cohomology, with supporting computations for Breuil–Kisin modules in Appendix A.

Significance. If the main theorem is correct, it is a significant advance: it provides an integral comparison with coefficients for arbitrary ramification degree e, going beyond the unramified case treated in earlier work such as [IKY25], and it recovers the classical Berthelot–Ogus isomorphism after rationalisation. The paper does not assume the conclusion or fit parameters; the ramified case is genuinely new. The explicit length calculations in Appendix A and the conjectural torsion framework in Section 4.4 are likely to be useful independently of the main proof. The central construction—twisting by Frobenius on the prismatisation of O_K as a prismatic analogue of Dwork's trick—is conceptually appealing and well explained. The main weakness is that several coherence statements in the proof are asserted rather than proved; these are internal proof-completeness gaps rather than contradictions with known results.

major comments (2)
  1. [Construction 3.19 and Proposition 3.23] The 'crystal property' invoked in Construction 3.19 and again in the proof of Proposition 3.23 (diagram (3.11)) is neither stated precisely nor proved, and no reference is given for it. This property is load-bearing: it is what identifies the value V(Sπ,π′, Iπ,π′) with the derived base changes of V(S, Eπ) along the two projections, and it must ensure that the two routes through diagram (3.11) induce a well-defined isomorphism independent of the uniformiser π. Please state the required coherence statement explicitly—ideally as a consequence of Proposition 2.5 together with the functoriality of pullback on X^Δ—and verify it for the non-flat maps appearing in (3.11) (for instance the reductions modulo u and the maps through (A_crys, p)). Without this, the naturality of the comparison in Theorem 4.9 is not established as stated.
  2. [Proposition 3.25, proof] The proof of Proposition 3.25 asserts that 'the right two trapeziums are 2-commutative with obvious identifications of the compositions.' This 2-commutativity is precisely what identifies the two composite maps from Spf(OK) to O^Δ_K that define the upper and lower routes, and it underlies Proposition 3.24, which compares the stack-theoretic isomorphism ι^(n) with the Breuil–Kisin-theoretic ι′^(n). The assertion is not obvious because the diagram involves non-flat maps (for example the section i : Spf(OK) → Spf(˜S) and the reduction maps). Please provide the explicit homotopies or give a complete verification of the 2-commutativity of both trapeziums.
minor comments (6)
  1. [Section 4.4, paragraph after Conjecture 4.19] The sentence 'Conjecture 4.26 can be understood as giving a rough relationship between the size of ai and the size of bj' should refer to Conjecture 4.19, not Conjecture 4.26.
  2. [Example 4.21] In the displayed computation, 'H3_crys(X/OK) = k' should be 'H3_crys(X_k/W) = k', and later in the paragraph 'H2_crys(X/OK)' should be 'H2_crys(X_k/W)'; the notation in Section 4.1 consistently uses H^i_crys(X_k/W, –).
  3. [Lemma A.13, proof] The proof says 'combining (A.9) and Lemma A.10' but the needed isomorphism (M^(n)_tor[E] ≃ M^(n)[E]) is Lemma A.12, not Lemma A.10.
  4. [Lemma A.17, proof] The phrase 'using (A.12) of Lemma A.11' should refer to Lemma A.13, since the short exact sequence labelled (A.12) is stated in Lemma A.13.
  5. [Proposition 3.23, diagram (3.11)] The object denoted (A_inf, φ^n(ξ)) is elsewhere written (A_inf, ξ^(n+1)); please use consistent notation for the pair (A_inf, φ^n(ξ^(1))).
  6. [Proof of Proposition 2.13] In the sentence 'This clearly happens if and only if p divides up^{n+1} in ˜S', the expression should read u^{p^{n+1}} (with the exponent p^{n+1} on u), as indicated by the subsequent inequality p^{n+1} ≥ p˜e.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: Theorem 4.9 follows from the stacky comparison Theorem 3.12 via pullback functors; the only same-author citations are unramified base lemmas, and the unproved crystal property is a proof gap, not a circular step.

full rationale

The central isomorphism is not obtained by assuming its conclusion. Theorem 4.9 is the functorial consequence of Theorem 3.12 applied to Rf^Δ_*V, and Theorem 3.12 is proved by comparing the explicit Cartier-Witt divisor and OK-structure maps of the twisted de Rham and crystalline points; no parameter is fitted and no torsion conjecture is used to establish it. The uniformiser-independence in Proposition 3.23 and Construction 3.19 does appeal to an unproved 'crystal property' for perfect complexes on the mixed Breuil-Kisin prism, but that property is a coherence statement of the prismatic formalism, not an assumption equivalent to the desired comparison, so it is a proof-completeness gap rather than circularity. The only citations overlapping with the present authors are [IKY25, Proposition 1.15 and Theorem 1.19], which supply unramified/W-level identifications used inside the proof of the stacky comparison; the ramified e>1 case is new, and these cited results are lemmas rather than the target isomorphism, so they are not load-bearing in the circular sense. Accordingly the derivation chain is self-contained up to standard prismatic formalism, and the conjectural torsion framework in Section 4.4 is explicitly not used for the main theorem.

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

The central claim rests on the prismatic and stacky foundations of Bhatt-Lurie and Drinfeld, on the perfectness theorem for relative prismatic pushforwards of Guo-Reinecke, and on standard divided-power facts. None of these are derived in the paper, which accounts for the axiom list. No numerical parameters are fitted; the threshold a is a definition, and the modified Breuil prism is an explicit mathematical construction, not an unexplained entity.

assumptions (7)
  • domain assumption The prismatisation functor and the equivalence D(X^Delta) is equivalent to D(X_Delta) exist and are compatible with pushforwards.
    Used throughout Sections 2 and 3; relies on [BL22a, BL22b, BS22, Dri24] and Proposition 2.5.
  • domain assumption Rf^Delta_* preserves perfect complexes and Frobenius structures for smooth proper morphisms of smooth formal O_K-schemes.
    Quoted as Theorem 2.7 from [GR24, Corollary 5.16 and Theorem 8.1]; needed to define twisted cohomology in Section 4.
  • domain assumption The map rho_S: Spf(S) to O_K^Delta is a flat surjection.
    Proposition 2.8; the proof uses [BL22b, Lemma 6.3] and the quasi-syntomicity of O_K to O_C.
  • domain assumption The crystal property for perfect complexes of prismatic crystals on the absolute prismatic site is valid in the derived sense.
    Invoked without proof in Construction 3.19 and the proof of Proposition 3.23 to identify pullbacks of V along the two projections of the mixed Breuil-Kisin prism.
  • domain assumption Derived base change for formal stacks holds for the cartesian diagrams used in Proposition 4.5.
    Proposition 4.5 relies on [Hau24, Proposition A.0.2] and the verification that rho_dR is a locally closed regular immersion.
  • standard math The divided powers pi^m/m! lie in O_K for all m.
    Used in Proposition 2.11 and the definition of the modified Breuil prism; follows from [BO83, Lemma 3.9].
  • domain assumption The identification of the de Rham point rho_dR with rho_S^(1) composed with nat (Proposition 3.10) holds for ramified O_K.
    Quoted from [IKY25, Proposition 1.15]; it underlies Corollary 3.11 and thus the twisted de Rham realisation.

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Pith. "Pith review of An integral comparison of crystalline and de Rham cohomology." pith.science (2026). https://pith.science/paper/YACTXLL5

@misc{pith2026250717631,
  author       = {Pith},
  title        = {Pith review of: An integral comparison of crystalline and de Rham cohomology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YACTXLL5}},
  note         = {Machine review of arXiv:2507.17631}
}
abstract

Let $\mathcal{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$ and fraction field $K$. It is a celebrated result of Berthelot and Ogus that for a smooth proper formal scheme $X/\mathcal{O}_K$ there exists a comparison between the de Rham cohomology groups $\mathrm{H}^i_\mathrm{dR}(X/\mathcal{O}_K)$ and the crystalline cohomology groups $\mathrm{H}^i_\mathrm{crys}(X_k/W(k))$ of the special fibre, after tensoring with $K$. In this article, we use the stacky perspective on prismatic cohomology, due to Drinfeld and Bhatt--Lurie, to give a version of this comparison result with coefficients in a perfect complex of prismatic $F$-crystals on $X$. Our method is of an integral nature and suggests new tools to understand the relationship between torsion in de Rham and crystalline cohomology.

Discussion (0). Continue with ORCID to comment.

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