{"id":"45c23267-ae17-4d8c-930b-871969ee0a07","arxiv_id":"1908.06366","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new construction of the crystalline-to-A_inf comparison map reproduces the crystalline, B_cris, and filtration comparisons for smooth proper formal schemes of good reduction.","lead":"This paper gives a new, simpler route to a central comparison in integral p-adic Hodge theory, matching crystalline cohomology with p-adic etale cohomology for schemes with good reduction. It builds a Frobenius-equivariant map from crystalline cohomology into A_inf-cohomology with A_cris coefficients, using descent to specially nice local rings.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Local PD-thickening lemmas 2.6/2.7 are the load-bearing point; their proofs are too abbreviated, and a failure in the p-th-power divisibility step would invalidate h_crys.","rationale":"The reader's weakest-assumption analysis identifies the same local lemmas that I find most load-bearing. The global construction of h_crys is a straightforward descent once Lemma 2.7 is granted, and the derivation of Theorems 4, 4.13, and 5.14 from the existence of h_crys is coherent. The real risk is concentrated in the two local statements: topological freeness of AΩ_S⊗A_cris and the PD-thickening property of the projection to S/p. The proof of Lemma 2.6 is too compressed to be machine-checked, and Lemma 2.7's proof relies on a chain of operations — φ^{-1}(ξ), p^{1/p}, and the vanishing of p-th powers in LΩ(S/p) — whose validity is tied exactly to the quasi-regular semiperfectoid hypotheses. The paper itself points to an omitted explicit computation in Remark 2.8, and Remark 3.8 records a further open issue about whether h_crys is an isomorphism. Since the reader already set CONDITIONAL on fuller details of these local lemmas, my read does not move the verdict. A concrete computational check on the canonical example is the most direct way to settle the concern.","tokens_in":20834,"tokens_out":32739,"duration_ms":342858,"concrete_test":"Carry out the explicit computation in Remark 2.8 for the model example S0 = O_C⟨X^{1/p^∞}⟩/(X) and its perfectoid variant: compute AΩ_{S0} and AΩ_{S0}⊗A_cris explicitly, verify topological freeness over A_cris, and verify that every element of ker(AΩ_{S0}⊗A_cris→S0) has p-th power divisible by p. If this computation cannot be completed or the kernel is not PD, Lemma 2.7 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.6 and Lemma 2.7 are the load-bearing local input. Lemma 2.6 asserts AΩ_S ⊗^L A_cris is topologically free and concentrated in degree 0; its proof is three lines and uses that LΩ(S/p) is free and concentrated in degree 0, which is exactly the nontrivial quasi-regularity input. Lemma 2.7 then needs this discreteness to identify AΩ_S⊗A_cris as an ordinary ring and to reduce the PD property to the p-th power criterion. The p-th power argument itself uses several hidden aids: the existence of the element φ^{-1}(ξ) in AΩ_S, surjectivity of θ to S, the operation 'p^{1/p}' as an element of S, and the implication α(x)=0 ⇒ x^p=0 in LΩ(S/p). These are plausible for quasi-regular semiperfectoid S, but they are not fully justified and are acknowledged as an omitted explicit computation in Remark 2.8. If any of these fail for a legitimate S∈qrsPerfd^proj, Section 3.1 cannot construct h_crys, and Theorem 1 collapses.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new construction of the crystalline comparison for A_inf-cohomology of smooth formal schemes over O_C in the case of good reduction. The main object is a functorial phi-equivariant map h_crys: RGamma_crys((X_O_C/p)/A_cris) -> RGamma_Ainf(X) tensor^L_{A_inf} A_cris, compatible with the de Rham comparison after base change (Theorem 1 and Theorem 3.6). The proof reduces via quasisyntomic descent to quasiregular semiperfectoid algebras S, for which it must show that AOmega_S tensor^L A_cris is a discrete topologically free A_cris-algebra (Lemma 2.6) and that its projection to S/p is a PD-thickening (Lemma 2.7). From this map the paper derives the crystalline specialization to W(k), the Breuil-Kisin-Fargues structure on H^i_Ainf(X), the B_cris comparison for Galois representations, and, via a new infinitesimal B_dR^+ site in Section 5, the filtration compatibility and some integral recovery results. The paper explicitly takes the de Rham comparison as an input and therefore does not aim to reprove that half of [BMS18].","tokens_in":21038,"tokens_out":13837,"duration_ms":136274,"significance":"If the local lemmas are correct, the paper gives a genuinely simpler route to the main comparison theorems of [BMS18], avoids relative de Rham-Witt machinery, and packages the B_dR^+ side in a way adapted to filtration statements. The direction of h_crys is well chosen for Galois-invariant arguments, and the derivation of Corollaries 3.7 and 4.7 and of Theorem 4.13 from a single functorial map is clean. The paper is transparent about its inputs and about the omitted local computation in Remark 2.8; there is no fitting and no circularity, since the target comparison is not used to prove itself. The significance is contingent on completing the proofs of Lemmas 2.6 and 2.7, which are the load-bearing local statements.","major_comments":[{"comment":"The proof is too abbreviated for a load-bearing statement. It asserts that LOmega(S/p) is a free O_C/p-module concentrated in degree 0 by 'considering the graded pieces wedge^i L(S/p)[-i] for the conjugate filtration', and that the D(A_inf)-valued sheaf R |-> AOmega_R 'takes discrete values on quasi-regular semiperfectoid objects'. The second assertion is essentially the discreteness half of what has to be proved, and the first does not by itself give a basis of the derived (p,xi)-completion AOmega_S. One needs an argument that the chosen lifts form a topological basis and that no higher Tor contributes when passing to AOmega_S tensor^L A_cris. Since Lemma 2.7 and Construction 3.1 use AOmega_S tensor^L A_cris as an ordinary ring concentrated in degree 0, this gap directly affects the existence of h_crys.","section":"§2.4, Lemma 2.6"},{"comment":"The verification that beta is a PD-thickening is the core local computation, but the proof is a sketch. After 'ker(can) is generated by xi' the conclusion 'x in (phi^{-1}(xi)) subset AOmega_S/(p,xi)' refers to the wrong quotient: the diagram works in AOmega_S/(p,xi^p). The subsequent expression x = phi^{-1}(xi)y + pz' + xi w and the substitution y = p^{(p-1)/p}y' + x_1 treat elements of S and of AOmega_S interchangeably and use phi^{-1}(xi) without specifying how this element is obtained. The 'repeat this procedure' step is not formalized, and convergence or termination of the iteration is not discussed. Remark 2.8 acknowledges that the explicit computation is omitted, but for the construction of h_crys this computation is exactly what must be supplied.","section":"§2.4, Lemma 2.7 and Remark 2.8"},{"comment":"The comparison between h_crys and h_can after base change reduces to a polynomial algebra and then to a perfectoid S. The proof says that by [BdJ11] the two maps agree, but the independence of the choice of the lift Sigma -> tilde{S}/p is not shown. If different lifts give different maps, the resulting h_S is not functorial and the limit in Construction 3.4 would be ill-defined. The proof should spell out why the crystalline-site construction makes h_S independent of auxiliary choices, or should give a canonical construction that does not depend on such choices.","section":"§3.1, Lemma 3.2"}],"minor_comments":[{"comment":"There is a typo in the paragraph before Theorem 1: 'our first first main result' should read 'our first main result'.","section":"§1.2"},{"comment":"The vertical map labelled 'omega |-> omega^p' and the implication alpha(x)=0 implies x^p=0 in LOmega(S/p) need a reference or a one-line justification; both are used without comment.","section":"§2.4, diagram in Lemma 2.7"},{"comment":"The identification RGamma_Ainf(X) tensor^L A_cris is isomorphic to the homotopy limit of AOmega_R tensor^L A_cris over affine opens Spf R of X_et is asserted without spelling out the required etale descent for the A_inf-cohomology sheaf; a brief justification or reference should be added.","section":"§3.2, Construction 3.4"},{"comment":"The notation B_cris^+ is used before it is defined in the text; it should be introduced explicitly, or a standard reference for the notation should be given.","section":"§4.3, Lemma 4.6"},{"comment":"The phrase 'the derived quotient Sigma_dR(.)/xi is isomorphic to Sigma_C(.)' would be clearer as a sentence about the termwise derived quotient of the cosimplicial ring Sigma_dR(.), since the individual terms are not all flat over B_dR^+.","section":"§5.2, Lemma 5.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the overall strategy is attractive. The main risk is the unproved local PD computation in Lemmas 2.6 and 2.7, which the author explicitly acknowledges in Remark 2.8; the surrounding framework and the cited results from [BMS19] and [Mor16] make a fix plausible, so I would not reject on the basis of the current gaps, but they must be filled before the manuscript can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is not a new comparison theorem. It is a simplified reproof of results already in [BMS18], and the paper says so plainly. The genuinely new object is the functorial Frobenius-equivariant map h_crys from RΓ_crys((X_O_C/p)/A_cris) to RΓ_Ainf(X)⊗^L A_cris, built by quasisyntomic descent. That map lets the author recover the crystalline specialization, the B_cris comparison, the Breuil–Kisin–Fargues structure, and filtration compatibility without the all-possible-coordinates machinery.\n\nThe paper does several things well. The introduction frames the contribution honestly: the final theorems are reproofs, and the de Rham comparison is taken as input—the easier half of [BMS18]. The construction of h_crys is coherent, and the deduction of the main theorems is efficient and mostly transparent. Lemma 3.2 cleanly reduces compatibility with the de Rham comparison to [BdJ11], and the infinitesimal-site reformulation of B_dR-cohomology in Section 5 is a useful addition. Citations look appropriate; there is no circularity and no fitting.\n\nThe real soft spot is the local input, Lemmas 2.6 and 2.7. Lemma 2.6 asserts topological freeness of AΩ_S⊗^L A_cris in three lines, and it leans on the same quasi-regularity input (discreteness and freeness of LΩ(S/p)) plus an implicit topological-basis lift. Lemma 2.7's p-th power divisibility step uses ϕ^{-1}(ξ), p^{1/p}, and the implication α(x)=0 ⇒ x^p=0 in LΩ(S/p). The proof is more detailed than the stress-test note suggests, but it still contains hidden aids, and Remark 2.8 acknowledges that the computation is omitted. If either lemma fails, h_crys does not exist and Theorem 1 collapses. I do not see a concrete reason to think they fail—the claims are plausible for qrsPerfd^proj—but the exposition is too terse for the load they carry.\n\nWho is this for? Specialists in integral p-adic Hodge theory who want a shorter route to the BMS comparisons or a template for the semistable case. It is not a breakthrough, but it is a genuine, useful simplification. My recommendation: send it to peer review, with the expectation that the referee asks for fuller proofs of Lemmas 2.6 and 2.7. A serious referee can check the local lemmas in finite time, and the rest of the paper will likely hold up.","headline":"A clearly written re-proof of known BMS comparison theorems via a new functorial crystalline map h_crys; the load-bearing local lemmas are compressed, but the paper deserves refereeing.","tokens_in":21615,"tokens_out":2318,"would_cite":true,"duration_ms":25373,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F30","14F40","14G22"],"pacs":[],"model":"deepseek-v4-flash","headline":"A canonical Frobenius-compatible map links crystalline cohomology to A_inf-cohomology for formal schemes with good reduction.","keywords":["A_inf-cohomology","crystalline cohomology","integral p-adic Hodge theory","quasiregular semiperfectoid rings","quasisyntomic descent","Breuil-Kisin-Fargues modules","B_cris comparison","good reduction"],"falsifier":"Compute the divided-power divisibility directly on the explicit ring $S_0=O_C\\langle X_j^{1/p^\\infty}\\rangle_{j\\in J}/(X_j)$: take $x$ in the kernel of $A\\Omega_{S_0}\\otimes^{\\mathbf L}A_{\\mathrm{cris}}\\to S_0$ corresponding to $X_j$ and check whether $x^p$ lies in $p$ times that kernel. A single $j$ where the divisibility fails falsifies Lemma 2.7 and with it the local construction of Section 3.1.","tokens_in":20576,"feed_emoji":"🧊","tokens_out":16295,"duration_ms":138183,"temperature":0.7,"pith_summary":"Integral p-adic Hodge theory studies the $A_{\\mathrm{inf}}$-cohomology $\\mathrm{R}\\Gamma_{A_{\\mathrm{inf}}}(X)$ of a smooth formal scheme over $O_C$, where $A_{\\mathrm{inf}}=W(O_C^\\flat)$ is the period ring built from the tilt of $O_C$ and $A_{\\mathrm{cris}}$ is its divided-power envelope. This paper explains that object crystallinely: it constructs a canonical Frobenius-equivariant map from the crystalline cohomology of $X_{O_C}/p$ over $A_{\\mathrm{cris}}$ into $\\mathrm{R}\\Gamma_{A_{\\mathrm{inf}}}(X)\\otimes^{\\mathbf L}_{A_{\\mathrm{inf}}}A_{\\mathrm{cris}}$. The map is assembled locally on quasiregular semiperfectoid covers, ring-theoretic analogues of local complete intersections, where the relevant $A_{\\mathrm{cris}}$-algebra is topologically free and the projection to $S/p$ carries divided powers. The result matters because the existence of this single map, in the direction from crystalline to $A_{\\mathrm{inf}}$-cohomology, is enough to reprove the principal comparison theorems of the area: crystalline specialization, the $B_{\\mathrm{cris}}$ comparison, the Breuil-Kisin-Fargues structure of the cohomology groups, and compatibility with Hodge filtrations, without the heavier relative de Rham-Witt and all-coordinates machinery.","feed_headline":"One map recovers the crystalline side of A-inf cohomology","feed_subtitle":"A local divided-power argument on semiperfectoid rings reproves B_cris comparison and Breuil-Kisin-Fargues structure.","key_machinery":"The load-bearing local object is the class of quasiregular semiperfectoid $O_C$-algebras $S$: p-torsion-free quotients of perfectoid rings by quasiregular ideals, with examples such as $O_C\\langle X^{1/p^\\infty}\\rangle/(X)$. For such $S$, Lemmas 2.6 and 2.7 show that $A\\Omega_S\\otimes^{\\mathbf L}A_{\\mathrm{cris}}$ is a topologically free $A_{\\mathrm{cris}}$-module concentrated in degree $0$, and that the natural projection $\\beta:A\\Omega_S\\otimes^{\\mathbf L}A_{\\mathrm{cris}}\\to S/p$ is a PD-thickening, meaning its kernel carries divided powers. This makes $A\\Omega_S\\otimes^{\\mathbf L}A_{\\mathrm{cris}}$ an object of the crystalline site of $R/p$ over $(A_{\\mathrm{cris}},O_C/p,\\gamma)$, so restriction along $R\\to S$ produces a canonical map from $\\mathrm{R}\\Gamma_{\\mathrm{crys}}((R/p)/A_{\\mathrm{cris}})$ to $A\\Omega_S\\otimes^{\\mathbf L}A_{\\mathrm{cris}}$; quasisyntomic descent over such $S$, followed by homotopy limits over affine opens of $X_{\\mathrm{et}}$, assembles these local maps into the global $h_{\\mathrm{crys}}$.","core_discovery":"Theorem 1 of the paper asserts that for a smooth formal scheme $X$ over $\\mathrm{Spf}\\,O_C$ there is a functorial $\\varphi$-equivariant map $h_{\\mathrm{crys}}:\\mathrm{R}\\Gamma_{\\mathrm{crys}}((X_{O_C}/p)/A_{\\mathrm{cris}})\\to \\mathrm{R}\\Gamma_{A_{\\mathrm{inf}}}(X)\\otimes^{\\mathbf L}_{A_{\\mathrm{inf}}}A_{\\mathrm{cris}}$ that becomes the inverse of the de Rham comparison $\\gamma_{\\mathrm{dR}}$ after base change to $O_C/p$. The direction is opposite to the map previously obtained by taking limits over coordinate choices, and the paper shows that existence alone is sufficient: derived Nakayama's lemma yields the crystalline specialization $\\mathrm{R}\\Gamma_{A_{\\mathrm{inf}}}(X)\\otimes^{\\mathbf L}_{A_{\\mathrm{inf}},\\vartheta}W(k)\\simeq \\mathrm{R}\\Gamma_{\\mathrm{crys}}(X_k/W(k))$; after base change to $B_{\\mathrm{cris}}$ and composition with the etale comparison one obtains the $B_{\\mathrm{cris}}$ comparison isomorphism; and a descending induction on the cohomological degree, using the equality of $W(k)$-rank and $\\mathbb Z_p$-rank, proves that each $\\mathrm{H}^i_{A_{\\mathrm{inf}}}(X)$ is a Breuil-Kisin-Fargues module (a finitely presented $A_{\\mathrm{inf}}$-module whose Frobenius becomes an isomorphism after inverting $\\xi$ and which is free after inverting $p$). A $B_{\\mathrm{dR}}^+$-version, formulated through an infinitesimal site on the generic fibre, supplies the filtration compatibility of the comparison.","pith_inferences":["Because the global map is assembled purely from local divided-power data, the whole comparison can be audited object-by-object: verifying the PD-thickening statement on the explicit rings $O_C\\langle X_j^{1/p^\\infty}\\rangle/(X_j)$ would certify the engine of the proof independently of the surrounding descent formalism.","Remark 3.8 leaves open whether $h_{\\mathrm{crys}}$ is itself a quasi-isomorphism; if it were, the missing $A_{\\mathrm{cris}}$-specialization statement would follow, but the non-finite-generation of $\\ker(A_{\\mathrm{cris}}\\to O_C/p)$ blocks the obvious derived Nakayama argument.","The author suggests the construction adapts to logarithmic settings; if the local PD-thickening lemma survives that variant, the same quasisyntomic-descent assembly would plausibly carry the comparison theorems to semistable reduction without a new global analysis."],"forward_implications":["The crystalline comparison $\\mathrm{R}\\Gamma_{A_{\\mathrm{inf}}}(X)\\otimes^{\\mathbf L}_{A_{\\mathrm{inf}},\\vartheta}W(k)\\simeq\\mathrm{R}\\Gamma_{\\mathrm{crys}}(X_k/W(k))$ follows by derived Nakayama's lemma once $h_{\\mathrm{crys}}$ exists.","The $B_{\\mathrm{cris}}$ comparison gives a $(\\mathrm{Gal}_K,\\varphi)$-equivariant isomorphism $\\mathrm{H}^i_{\\mathrm{crys}}(X_{k_0}/W(k_0))\\otimes B_{\\mathrm{cris}}\\simeq\\mathrm{H}^i_{\\mathrm{et}}(X_C^{\\mathrm{ad}},\\mathbb Z_p)\\otimes B_{\\mathrm{cris}}$, so $\\mathrm{H}^i_{\\mathrm{et}}(X,\\mathbb Q_p)$ is a crystalline Galois representation.","Each group $\\mathrm{H}^i_{A_{\\mathrm{inf}}}(X)$ is a Breuil-Kisin-Fargues module, and the cohomology vanishes for $i>2\\dim X/O_C$.","The torsion inequalities $\\mathrm{length}_{W(k)}(\\mathrm{H}^i_{\\mathrm{crys}}(X_k/W(k))_{\\mathrm{tor}}/p^n)\\ge\\mathrm{length}_{\\mathbb Z_p}(\\mathrm{H}^i_{\\mathrm{et}}(X,\\mathbb Z_p)_{\\mathrm{tor}}/p^n)$ hold for every $n\\ge 1$.","When $\\mathrm{H}^i_{\\mathrm{crys}}(X_k/W(k))$ and $\\mathrm{H}^{i+1}_{\\mathrm{crys}}(X_k/W(k))$ are torsion free, the integral crystalline cohomology with its Frobenius action is recovered from the etale cohomology together with the $B_{\\mathrm{dR}}^+$-lattice $\\mathrm{H}^i_{\\mathrm{inf}}(X/B_{\\mathrm{dR}}^+)$."],"supporting_citations":[{"why":"defines $\\mathrm{R}\\Gamma_{A_{\\mathrm{inf}}}(X)$, supplies the etale and de Rham comparisons taken as input, and states the comparison theorems being reproved.","marker":"[BMS18]"},{"why":"supplies the quasisyntomic site, quasiregular semiperfectoid rings, and the basis and descent facts used to localize and assemble $h_{\\mathrm{crys}}$.","marker":"[BMS19]"},{"why":"provides the descending induction and module lemmas used to show the cohomology groups are Breuil-Kisin-Fargues modules.","marker":"[Mor16]"},{"why":"supplies the pro-etale $B_{\\mathrm{dR}}^+$ period sheaf, the etale comparison, and the $B_{\\mathrm{dR}}^+$-Poincare lemma used for filtration compatibility.","marker":"[Sch13]"},{"why":"provides the polynomial-ring comparison and formal Poincare lemma used in the local proof that $h_S$ agrees with the de Rham map.","marker":"[BdJ11]"},{"why":"offers an alternative derivation of the de Rham comparison that the paper can use as input.","marker":"[Bha17]"},{"why":"gives the earlier crystalline-to-pro-etale map that motivates but cannot recover $h_{\\mathrm{crys}}$, explaining why the new direction is needed.","marker":"[Fal02]"}],"fun_headline_variants":["One map flips direction, proves Ainf-crystalline comparison","Crystalline comparison via a single divided-power map","Existence alone yields Ainf-crystalline isomorphism","Ainf-crystalline bridge from a derived Nakayama step","Reproves B_cris comparison with one Ainf map"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the local lemma that for every p-torsion-free quasiregular semiperfectoid $O_C$-algebra $S$, the projection $A\\Omega_S\\otimes^{\\mathbf L}A_{\\mathrm{cris}}\\to S/p$ is a divided-power thickening; if that divisibility statement fails, the map $h_{\\mathrm{crys}}$ cannot be assembled.","fun_headline_variants_meta":{"raw":{"variants":["One map flips direction, proves Ainf-crystalline comparison","Crystalline comparison via a single divided-power map","Existence alone yields Ainf-crystalline isomorphism","Ainf-crystalline bridge from a derived Nakayama step","Reproves B_cris comparison with one Ainf map"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000656,"raw_usage":{"total_tokens":2996,"prompt_tokens":932,"completion_tokens":2064,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":1985}},"tokens_in":548,"tokens_out":2064,"duration_ms":14981,"temperature":1.0,"reasoning_tokens":1985,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:48:05.478583+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the divided-power divisibility directly on the explicit ring $S_0=O_C\\langle X_j^{1/p^\\infty}\\rangle_{j\\in J}/(X_j)$: take $x$ in the kernel of $A\\Omega_{S_0}\\otimes^{\\mathbf L}A_{\\mathrm{cris}}\\to S_0$ corresponding to $X_j$ and check whether $x^p$ lies in $p$ times that kernel. A single $j$ where the divisibility fails falsifies Lemma 2.7 and with it the local construction of Section 3.1.","supporting_citations":[],"review_version":1}