{"id":"a93e891a-9f90-4988-8127-53df54275a19","arxiv_id":"2507.17631","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves an integral, coefficient-version of the Berthelot-Ogus comparison between de Rham and crystalline cohomology using stacky prismatic cohomology and a Dwork-trick argument.","lead":"Mathematicians prove a new integral version of a classical comparison between crystalline and de Rham cohomology for schemes over ramified p-adic rings, now also with coefficients. The result gives a new framework for relating torsion in the two cohomology theories, a subtle question in arithmetic geometry.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop. 3.23's uniformiser-independence is asserted via an unproved 'crystal property' for perfect complexes; Theorem 4.9's naturality depends on it.","rationale":"The reader's weakest_assumption already isolates the crystal property for perfect complexes in Construction 3.19 and Proposition 3.23. My reading confirms that this is the point where Theorem 4.9's naturality and uniformiser-independence are decided. Proposition 2.5 is a strong foundational result, but the paper does not spell out how it yields the triangulated coherence used in diagram (3.11); the proof of Proposition 3.23 is a single sentence. This is an internal completeness issue, not a dispute with the expected comparison: the stack-theoretic theorem and the Breuil–Kisin calculations around it are coherent, and the appendix supports the torsion framework. I found no actual contradiction in the stated results, so the appropriate posture remains a low-confidence conditional: the central claim is plausible but not fully verified as written. Hence no verdict change.","tokens_in":34594,"tokens_out":38694,"duration_ms":432654,"concrete_test":"Derive from Proposition 2.5 the following coherence lemma: for every V in Perf((O_K)^Delta), the two base-change maps phi^{*(n+1)}_S V(S,E_pi) ⊗_S A_crys → phi^{*(n+1)}_{A_crys} V(A_crys,p) ← phi^{*(n+1)}_S V(S,E_pi') ⊗_S A_crys coincide after identifying both targets with V(A_crys,p). Test the lemma on a non-free perfect complex, e.g. V with V(S,E_pi)=S/(E_pi) and p=3, e=2, n=0, and then on a u-torsion Breuil–Kisin module from Appendix A. A mismatch for any such V would make the pi-independence in Proposition 3.23 false; a proof would close the gap in Theorem 4.9.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the passage from the stack-theoretic comparison (Theorem 3.12) to a comparison that is independent of the uniformiser and compatible with the Breuil–Kisin realisations. Construction 3.19 and the proof of Proposition 3.23 both appeal to the 'crystal property' for perfect complexes: for V in Perf((O_K)^Delta), the value on the mixed Breuil–Kisin prism S_{pi,pi'} is identified with the base change of V(S,E_pi) along either projection, and the two identifications are compatible. Proposition 2.5 gives an equivalence D(X^Delta) ≃ lim_{(A,I)} D(A), but the paper does not extract from it the precise coherence statement needed for diagram (3.11), which involves non-flat maps and the two 'u=0' routes through (W,p), (tilde S_pi,p), (A_crys,p), (S,E_pi^(1)), (A_inf, phi^n(xi)). If the two routes give non-identical equivalences, the isomorphisms iota'^{(n)}_pi depend on pi, the identifications psi^{(n)}_crys and psi^{(n)}_dR do not conjugate iota^{(n)} to iota'^{(n)}, and the natural comparison in Theorem 4.9 is not established as stated. This is a proof-completeness gap, not a contradiction with the expected result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":34870,"tokens_out":10604,"duration_ms":107141,"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":[{"comment":"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.","section":"Construction 3.19 and Proposition 3.23"},{"comment":"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.","section":"Proposition 3.25, proof"}],"minor_comments":[{"comment":"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.","section":"Section 4.4, paragraph after Conjecture 4.19"},{"comment":"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, –).","section":"Example 4.21"},{"comment":"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.","section":"Lemma A.13, proof"},{"comment":"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.","section":"Lemma A.17, proof"},{"comment":"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))).","section":"Proposition 3.23, diagram (3.11)"},{"comment":"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.","section":"Proof of Proposition 2.13"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is plausible and the overall strategy is sound, but the proof as written leaves two load-bearing coherence statements at the level of sketches: the 'crystal property' for perfect complexes and the 2-commutativity of the trapeziums in Proposition 3.25. These gaps are likely fillable within the manuscript's scope, and the paper does not appear to assume the conclusion. The reliance on [IKY25] is legitimate because that work is limited to the unramified case; still, the authors should clearly delineate which parts of the argument are new. The paper fits the journal's scope and, if the gaps are addressed, would be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: this paper proves an integral, coefficient-version of Berthelot–Ogus for smooth proper formal O_K-schemes with arbitrary ramification, using the stacky prismatic formalism. The main theorem (4.9) is new: the unramified case is in Imai–Kato–Youcis, and Bhatt–Lurie had Spf(Z_p), but the e>1 case with perfect complexes of prismatic F-crystals is not in the literature. The strategy is also nice: Dwork's trick is reformulated as an identification of maps into O_K^Δ, and the paper gives both a stacky and a Breuil–Kisin based comparison.\n\nWhat the paper does well: the structure is clear, the stacky interpretation is genuinely illuminating, and the appendix contains real computations (length inequalities for u^∞-torsion in Breuil–Kisin cohomology) that support the torsion conjectures. The authors also explicitly thank someone for pointing out a mistake in an earlier version, which is a good sign.\n\nThe soft spots are real but not fatal. The proof of Proposition 3.23, which is needed for uniformiser-independence, invokes a 'crystal property' for perfect complexes to identify pullbacks along the two projections of the mixed Breuil–Kisin prism. This is not proved; it is asserted in Construction 3.19 and again in the proof of Proposition 3.23. The stress-test is right that this is a proof-completeness gap. The same goes for the 2-commutativity of the trapeziums in Proposition 3.25, which are dispatched as 'obvious'. A referee will need to see these spelled out. Also, the paper leans on unpublished notes of Bhatt and on [GL25]; that is not a flaw, but it makes independent verification harder.\n\nNone of this looks like a circular argument. The comparison is derived from the formalism, not from the conclusion. The reliance on [IKY25] is legitimate.\n\nBottom line: this deserves a serious referee. The main theorem is important if the gap is filled, and the framework for torsion is well-motivated. I'd send it out, but with a clear request that the crystal property be proved or precisely referenced.","headline":"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.","tokens_in":35446,"tokens_out":2238,"would_cite":true,"duration_ms":23348,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F30","14F40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["prismatic cohomology","crystalline cohomology","de Rham cohomology","Berthelot–Ogus comparison","integral p-adic Hodge theory","prismatic F-crystals","torsion in cohomology","Breuil–Kisin modules"],"falsifier":"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.","tokens_in":34374,"feed_emoji":"🧮","tokens_out":10024,"duration_ms":91141,"temperature":0.7,"pith_summary":"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.","feed_headline":"Twisted crystalline and de Rham cohomology match integrally","feed_subtitle":"A prismatic Dwork trick removes the need to invert p and opens a route to torsion comparisons.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $e \\leq p-1$."],"supporting_citations":[{"why":"The classical rational Berthelot–Ogus comparison which the paper extends to an integral statement with coefficients.","marker":"[BO83]"},{"why":"Provides the ambient prismatic cohomology formalism and the Breuil–Kisin prism used throughout.","marker":"[BS22]"},{"why":"Gives the stacky interpretation of prismatic cohomology and the identification of the Frobenius-twisted points used in the Dwork trick.","marker":"[BL22a]"},{"why":"Introduces the prismatization $X^\\Delta$ of $p$-adic formal schemes, the central stack.","marker":"[Dri24]"},{"why":"Supplies the equivalence between perfect prismatic crystals and crystalline crystals, and the pushforward theorem (Theorem 2.7) used to define the twisted realisations.","marker":"[GR24]"},{"why":"Establishes the unramified case and the identification of the de Rham point $\\rho_{\\mathrm{dR}}$ with $\\rho_S^{(1)}\\circ\\mathrm{nat}$ used in Proposition 3.10.","marker":"[IKY25]"},{"why":"Provides the structure theory of $u^\\infty$-torsion in Breuil–Kisin cohomology on which Conjecture 4.26 and the Appendix A arguments rest.","marker":"[LL23]"}],"fun_headline_variants":["Integral prismatic trick matches crystalline and de Rham","No p-inversion needed for crystalline-de Rham comparison","Perfect crystals get integral cohomology isomorphism","Stacky prismatic proof unifies cohomology integrally","Torsion-aware integral comparison via Dwork trick"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Integral prismatic trick matches crystalline and de Rham","No p-inversion needed for crystalline-de Rham comparison","Perfect crystals get integral cohomology isomorphism","Stacky prismatic proof unifies cohomology integrally","Torsion-aware integral comparison via Dwork trick"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000194,"raw_usage":{"total_tokens":1389,"prompt_tokens":1013,"completion_tokens":376,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":629,"completion_tokens_details":{"reasoning_tokens":298}},"tokens_in":629,"tokens_out":376,"duration_ms":4283,"temperature":1.0,"reasoning_tokens":298,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:45:45.780098+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}