{"id":"a5ede7dd-884e-4e3e-a287-eec5b3358a62","arxiv_id":"2506.11736","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a p-divisible group G over a smooth projective X/k in characteristic p, the formal group R^i f_fppf* G is isogenous to a p-divisible group whose Dieudonné crystal is the slope-[0,1] part of R^i f_crys* M^cr(G).","lead":"The paper proves that higher direct images of p-divisible groups along smooth projective varieties are p-divisible up to isogeny, and identifies their Dieudonné crystals with the slope [0,1] part of crystalline cohomology. This answers the rational form of a question of Artin and Mazur about enlarged formal Brauer groups, even over imperfect base fields.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.10 is false for the slope-1 crystal D_{1,1}; the m=n=1 step cannot hit the constant coefficient, so the proof of Theorem 3.11 has a genuine gap.","rationale":"The reader correctly spotted Proposition 3.10 as the weakest point, but the diagnosis given was an absent 'Lemma below' and an asserted extension of [22, Lemma 5.8(2)]. The stress-test pass finds a stronger, internal failure: Proposition 3.10 is false as stated for the simple slope-1 F-crystal D_{1,1}. Evaluating at R = k gives coker(phi - p) = W(k), of infinite p-exponent, because Frobenius acts trivially on W(k). The attempted fix in the m = n = 1 case ignores the constant coefficient of the input b; no etale extension can make a nonzero W(k)-constant into a sigma - 1 coboundary, since sigma fixes W(k). This is not a matter of differing from consensus; it is a direct algebraic counterexample inside the paper's own framework. The main theorem may still be true, and the issue may be repairable by treating slope 1 separately or by correcting the definition of the slope-[0,1] part, but the proof as written relies on the false Proposition 3.10 and therefore cannot be accepted in its current form. The missing internal lemma is real but secondary; it would not resolve the constant-term obstruction. For these reasons the reader's CONDITIONAL verdict should be strengthened to REJECT pending a corrected proof, rather than merely conditional acceptance.","tokens_in":13212,"tokens_out":22089,"duration_ms":235784,"concrete_test":"Recompute Proposition 3.10 for E = D_{1,1} at the regular semiperfect ring R = k (P = k, J = 0). Since sigma is the identity on W(k), verify that phi_E - p is the zero map and hence coker(u_*E)(Spec k) is isomorphic to W(k), which is not annihilated by any p^N. Then check the m = n = 1 equation p(sigma(x) - x) = p over every etale R'/k: the W(k)-component of sigma(x) - x is always 0, so no solution exists. If these computations confirm, Proposition 3.10 is false and the proof of Theorem 3.11 needs a separate argument for the slope-1 part.","verdict_should_be":"REJECT","load_bearing_attack":"Proposition 3.10 is the load-bearing input for the 'almost splitting' of the long exact sequence (3.3), and it is false as stated. Take E = D_{1,1} over S = Spec(k) with k perfect, i.e. a free W(k)-module with basis e and phi(e) = p e. Evaluating u_*E at R = k, one has A_crys(k) = W(k) and sigma is the identity on W(k), so phi_E - p = p(sigma - 1) is the zero map. Its cokernel is therefore W(k), which is not annihilated by any power of p. The proof's m = n = 1 case attempts to solve p(sigma(x) - x) = b by writing b = p(a0 + sum a_i gamma_i(f)) and inverting 1 - sigma on the divided-power part. This leaves the constant coefficient a0 untouched; passing to an etale R-algebra R' cannot help, because the W(k)-component of sigma(x) - x is zero for every x. Thus no etale cover kills nonzero constants in the cokernel. The absent 'Lemma below' would only affect the gamma_i(f) terms and does not repair the constant obstruction. Since slope-1 pieces such as the covariant Dieudonne module of mu_{p^infinity} are included in the slope-[0,1] part appearing in Theorem 3.11, the proof cannot go through Lemma 3.5 and Proposition 3.4 as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that for a smooth projective X over a field k finitely generated over a perfect field of characteristic p, and for a p-divisible group G over X, the higher direct image formal group R^i f_fppf* G is isogenous to a p-divisible group H^i, with a canonical isomorphism of F-isocrystals M_cr(H^i)_Q ≅ R^i f_crys* M^cr(G)_Q^{[0,1]}. The intended contribution is a rational answer to Artin-Mazur's question on enlarged formal Brauer groups, valid over imperfect fields. The proof proceeds by pushing forward the syntomic exact sequence of Trihan-Vauclair, proving an almost splitting of the resulting long exact sequence, and then using Dieudonné theory to identify the divisible part.","tokens_in":13513,"tokens_out":7422,"duration_ms":78233,"significance":"If the main theorem is correct, it is a substantial and natural generalization of the Artin-Mazur / Nygaard-Ogus description of formal Brauer groups: it identifies the rational Dieudonné module of the higher direct image formal group with the slope-[0,1] part of the crystalline Dieudonné F-isocrystal, and it proves p-divisibility up to isogeny. The paper is concise and the overall strategy is coherent, drawing on modern syntomic and crystalline methods. However, the proof as written has a concrete gap in the key almost-splitting proposition, and one essential existence input is imported from the literature by assertion rather than by proof. These issues are load-bearing for Theorem 3.11, so the manuscript needs nontrivial revision before it can be accepted.","major_comments":[{"comment":"The m=n=1 case of Proposition 3.10 is not proved. The text refers to a 'Lemma below' for the identity σγ_i(f)=p^i g γ_{pi}(f), but no such lemma appears in the manuscript. More importantly, the step 'we can find R' an étale R-algebra ... such that there exists x ∈ A_crys(R') and σ(x)-x=b' is asserted without justification; in particular, the constant W(k)-component of b is never addressed. For an algebraically closed k, surjectivity of σ-1 on W(k) would handle that component, but the proof does not say this. For a perfect but not algebraically closed field such as k=F_p, σ is the identity on W(k) and the constant part is an obstruction not killed by any power of p. Since Proposition 3.10 is exactly what makes the long exact sequence (3.3) split almost everywhere via Proposition 3.4, this gap is load-bearing for Theorem 3.11.","section":"§3.2, Proposition 3.10"},{"comment":"The existence of the p-divisible group H^i with M(H^i) isogenous to M^{[0,1]} is essential to the statement, but it is imported by the parenthetical assertion that [22, Lemma 5.8(2)] extends from smooth varieties to finitely generated fields, citing [8, Theorem 1]. The cited de Jong result is not the same as the needed statement, and the construction of H^i as an actual p-divisible group over S from the slope-[0,1] part of an F-crystal is not explained. Without a proof or a precise reference for this extension, the isomorphism M_cr(H^i)_Q ≅ M_Q^{[0,1]} is partly an input rather than a consequence of the argument.","section":"§3.3, proof of Theorem 3.11"},{"comment":"Definition 2.9 defines the slope-[0,1] part only in the isogeny category of F-isocrystals, but the proof of Theorem 3.11 uses maps such as M[0,1] -> M -> M' and diagram (3.4) as if they were maps of F-crystals. The passage from an isogeny-class construction to actual compatible maps of crystals needs to be justified, or the diagram must be reformulated in the isogeny category throughout.","section":"§2.2, Definition 2.9 and diagram (3.4)"}],"minor_comments":[{"comment":"The abbreviation 'al. isomorphism' is used frequently but never defined; the authors should define it as a map whose kernel and cokernel are annihilated by a fixed power of p, uniformly in the relevant parameters.","section":"General"},{"comment":"There are several typos and small errors: 'inducitve systems' in Lemma 2.4, 'simly write' in the Notations, 'Mitterage-Leffler' in Lemma 3.8, 'commutive diagram' in Remark 3.3, and 'founded' for 'funded' in the acknowledgments.","section":"Throughout"},{"comment":"In the proof of Lemma 3.1(1), the sentence 'By [15, Lemma 4.12], it suffices to prove the property on the big crystalline-Zariski site' is terse; a sentence explaining how this reduction works would help the reader.","section":"§3.1, Lemma 3.1"},{"comment":"The last isomorphism in Corollary 3.13 is justified only by 'follows from the proof of theorem 3.11'; this should be expanded or replaced by a precise citation to the step in the proof where slopes >1 are shown to be killed by φ-p.","section":"§3.3, Corollary 3.13"}],"recommendation":"major_revision","confidential_remarks":"The central claim is plausible and would be a nice result if the gap in Proposition 3.10 can be repaired. The missing 'Lemma below' and the unexplained constant-coefficient step are not mere presentation issues; they control the almost splitting of (3.3). The authors should be asked to supply the missing lemma and to give a real proof of the finitely generated field case of the input from [22]. If they can do so, the paper is likely publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves a real theorem: for a smooth projective X over a field k finitely generated over perfect, and for any p-divisible group G on X, the formal group R^i f_fppf* G is isogenous to a p-divisible group, and its Dieudonné crystal is the slope-[0,1] part of the crystalline F-isocrystal. This is a genuine advance over the known special cases (Oda for i=1, Nygaard–Ogus for K3, Grothendieck–Messing) and it answers the rational form of Artin–Mazur's question, even over imperfect bases. The strategy is sound: reduce to syntomic cohomology, use the slope filtration, then compare via the long exact sequence of Trihan–Vauclair. The paper is clearly written by people who know the machinery.\n\nThe main soft spot is Proposition 3.10, which is load-bearing. The proof of the m=n=1 case refers to a “Lemma below” that does not exist, and the step passing to an étale R' to handle the constant coefficient is asserted rather than shown. That is a genuine gap in the written proof. I also checked the stress-test note claiming Proposition 3.10 is false for the slope-1 crystal D_{1,1}. I don't think that counterexample lands: the proposition assumes k is algebraically closed, not F_p, and over an algebraically closed perfect field the map σ-1 on W(k) is surjective (the Artin–Schreier obstruction vanishes). So the missing lemma is likely just missing exposition, not a false proposition. But the referee should insist on a full write-up of that case.\n\nThe other weak point is the appeal to [22, Lemma 5.8(2)] for finitely generated fields; the paper says “the proof still works” but doesn't give the argument. That is probably fixable, but it is another spot where the reader has to take the authors' word. The self-citation to [15] is for a supporting lemma and is not problematic.\n\nThe central theorem is important and the overall architecture is convincing. The proof has two real but probably repairable gaps. A serious referee should be able to fill them or find what's missing. This deserves peer review, not a desk rejection. My advice: send it to a referee who knows crystalline Dieudonné theory and ask specifically about Proposition 3.10 and the [22] extension.","headline":"A genuinely new theorem—rational Artin-Mazur for higher direct images—with a plausible proof that has a couple of under-supported steps; worth refereeing.","tokens_in":14121,"tokens_out":4427,"would_cite":true,"duration_ms":46425,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F30","14L15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Higher direct images of p-divisible groups match slope-[0,1] crystals","keywords":["p-divisible groups","F-isocrystals","Dieudonné crystals","syntomic cohomology","formal Brauer groups","crystalline cohomology","Artin–Mazur question","slope filtration"],"falsifier":"Compute directly the cokernel of σ − p on A_crys(R) for a regular semiperfect ring such as R = k[$X^{{p^{-∞}}$}]/(X), where P = k[$X^{{p^{-∞}}$}] and J = (X); check whether every element of p A_crys(R) lies in the image of σ − p after multiplication by a fixed p^N. The proof of Proposition 3.10 says yes for all such R but leaves the computation to a missing lemma; finding one R where the cokernel is not killed by any fixed p^N would disprove the proposition and with it Theorem 3.11.","tokens_in":12970,"feed_emoji":"📐","tokens_out":9571,"duration_ms":87247,"temperature":0.7,"pith_summary":"This paper proves that, over a field of characteristic p, the higher direct images of a p-divisible group along a smooth projective morphism are, up to isogeny, p-divisible groups themselves. It identifies their Dieudonné crystals: the divisible part's crystal is canonically the slope-[0,1] part of the crystalline pushforward of the original Dieudonné crystal, as F-isocrystals. This settles the rational form of a question of Artin and Mazur about the enlarged formal Brauer group, and extends earlier results for K3 surfaces and for the first cohomology of μ_{p^∞} to all dimensions and to imperfect base fields. The method is to show that a pushforward of a syntomic exact sequence almost splits, giving almost-isomorphisms between the p^n-torsion of the higher direct image and kernels of Frobenius minus p in crystalline cohomology.","feed_headline":"Higher direct images of p-divisible groups match slope-[0,1] crystals","feed_subtitle":"Divisible part of R^i f_* G has Dieudonné module the slope-[0,1] crystalline pushforward. Answers Artin–Mazur rationally.","key_machinery":"The load-bearing object is the syntomic Dieudonné crystal M(G) and the exact sequence 0 → G[p^n] → $Fil^{1}$ M(G[p^n]) → M(G[p^n]) → 0 on the small syntomic site (Theorem 3.2 from [26]), which ties the p^n-torsion of G to the crystal via a divided Frobenius φ'. Pushing this sequence forward along f produces the long exact sequence (3.3). The paper proves this sequence almost splits by establishing (Proposition 3.10) that for any non-degenerate F-crystal E on the big crystalline-syntomic site of a perfect field, the map φ_E − p on u_*E has cokernel of finite exponent; this uniform bound makes the connecting maps in (3.3) almost zero. An isogeny step via a lemma of Pál realizes the slope-[0,1] part of the crystalline pushforward as the Dieudonné crystal of a p-divisible group H^i, and Lemma 2.4 upgrades the almost-isomorphisms to genuine isogenies between formal groups.","core_discovery":"The paper's central theorem (Theorem 3.11) asserts that for a smooth projective morphism f: X → Spec(k) with k a field finitely generated over a perfect field of characteristic p, and any p-divisible group G over X, the higher direct image formal group R^i f_fppf* G is isogenous to a p-divisible group H^i, and there is a natural isomorphism of F-isocrystals M_cr(H^i)_Q ≅ R^i f_crys* M^cr(G)$_Q^{{[0,1]}}$. Here M^cr(G) is the covariant Dieudonné crystal of G on the big crystalline-syntomic site, and the superscript [0,1] denotes the slope-[0,1] part in the isogeny category of F-crystals. This gives the rational form of Artin–Mazur's question for the enlarged formal Brauer group when G = μ_{p^∞} and i = 2, without assuming the base field is perfect.","pith_inferences":["If the missing computational lemma in Proposition 3.10 is supplied, the isogeny in Theorem 3.11 might be promoted to an isomorphism after a suitable modification of H^i, yielding an integral (not just rational) answer to Artin–Mazur's question.","Because the theorem works for any p-divisible group G, the same slope-[0,1] description should hold for the p^n-torsion sheaves of higher direct images of finite flat group schemes obtained by base change from G, such as μ_{p^n} and Z/p^n, with the same F-isocrystal.","The result suggests that the isogeny class of R^i f_* G is constant on Newton-strata where the relative crystalline slopes are constrained to [0,1]; on such strata the formal group is 'p-divisible up to isogeny' by a uniform mechanism.","A relative version over a smooth base scheme (not just a field) would follow if the perfect-field classification and the étale-extension argument in Proposition 3.10 can be replaced by a base-scheme statement; the current proof is field-specific."],"forward_implications":["The rational Artin–Mazur question is answered: for G = μ_{p^∞} and i = 2, the enlarged formal Brauer group's Dieudonné module is canonically the slope-[0,1] part of H^2_crys(X/W) ⊗ K, over any finitely generated field of characteristic p.","For every p-divisible group G, the isogeny class of the formal group R^i f_* G is completely determined by the F-isocrystal R^i f_crys* M^cr(G)_Q^{[0,1]}, and the formation is functorial.","The φ = p part of H^0_crys(S/W, R^i f_crys* M^cr(G)) computes the rational Tate module (lim H^0_fppf(S, R^i f_* G[p^n]))_Q, generalizing a prior result of Li–Qin with a cleaner proof.","The theorem extends the Nygaard–Ogus computation for K3 surfaces of finite height to arbitrary smooth projective varieties and all p-divisible groups, without requiring a lift to characteristic zero."],"supporting_citations":[{"why":"Supplies the syntomic complex and the exact sequence 0 → G[p^n] → Fil^1 M(G[p^n]) → M(G[p^n]) → 0 that is the starting point for the long exact sequence.","marker":"[26]"},{"why":"Poses the question answered and defines the enlarged functor Ψ and its relation to H^2_crys.","marker":"[1]"},{"why":"Proves representability of R^i f_* μ_{p^n} by affine finite-type group schemes, allowing passage from sheaves to formal groups.","marker":"[5]"},{"why":"Provides the F-crystal abelian category condition and previous methods for p-torsions of Brauer groups, used in Lemma 3.1.","marker":"[15]"},{"why":"Supplies the lemma that a slope-[0,1] F-crystal is isogenous to the Dieudonné crystal of a p-divisible group, giving H^i.","marker":"[22]"},{"why":"Provides the slope filtration of F-crystals, used to define the slope-[0,1] part and to show that φ_{M_{>1}} − p is an isomorphism.","marker":"[13]"},{"why":"Gives the perfect complex finiteness of crystalline cohomology, used to construct the F-crystal isogenous to R^i f_crys* M^cr(G).","marker":"[4]"}],"fun_headline_variants":["Higher direct images of p-divisible groups are isogenous to p-divisible groups","Dieudonné crystal of divisible part matches slope-[0,1] pushforward","Artin-Mazur rational question answered for higher direct images","F-isocrystals of higher direct images align with slope-[0,1] parts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on Proposition 3.10, whose m = n = 1 case invokes a 'Lemma below' that is not present in the text and asserts the existence of an étale extension R' solving an equation in A_crys(R); if the cokernel of φ_E − p is not killed by a fixed power of p, the almost-splitting of the long exact sequence fails and the main theorem collapses.","fun_headline_variants_meta":{"raw":{"variants":["Higher direct images of p-divisible groups are isogenous to p-divisible groups","Dieudonné crystal of divisible part matches slope-[0,1] pushforward","Artin-Mazur rational question answered for higher direct images","F-isocrystals of higher direct images align with slope-[0,1] parts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001019,"raw_usage":{"total_tokens":4276,"prompt_tokens":900,"completion_tokens":3376,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":3291}},"tokens_in":516,"tokens_out":3376,"duration_ms":23112,"temperature":1.0,"reasoning_tokens":3291,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:03:42.945345+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute directly the cokernel of σ − p on A_crys(R) for a regular semiperfect ring such as R = k[$X^{{p^{-∞}}$}]/(X), where P = k[$X^{{p^{-∞}}$}] and J = (X); check whether every element of p A_crys(R) lies in the image of σ − p after multiplication by a fixed p^N. The proof of Proposition 3.10 says yes for all such R but leaves the computation to a missing lemma; finding one R where the cokernel is not killed by any fixed p^N would disprove the proposition and with it Theorem 3.11.","supporting_citations":[{"cited_title":"A comparison theorem for semi-abelian schemes over a smooth curve","cited_arxiv_id":"1505.02942","evidence_quote":"Supplies the syntomic complex and the exact sequence 0 → G[p^n] → Fil^1 M(G[p^n]) → M(G[p^n]) → 0 that is the starting point for the long exact sequence."},{"cited_title":"Artin and B","cited_arxiv_id":null,"evidence_quote":"Poses the question answered and defines the enlarged functor Ψ and its relation to H^2_crys."},{"cited_title":"On p-torsions of geometric Brauer groups","cited_arxiv_id":"2406.19518","evidence_quote":"Provides the F-crystal abelian category condition and previous methods for p-torsions of Brauer groups, used in Lemma 3.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the lemma that a slope-[0,1] F-crystal is isogenous to the Dieudonné crystal of a p-divisible group, giving H^i."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the slope filtration of F-crystals, used to define the slope-[0,1] part and to show that φ_{M_{>1}} − p is an isomorphism."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the perfect complex finiteness of crystalline cohomology, used to construct the F-crystal isogenous to R^i f_crys* M^cr(G)."}],"review_version":1}