{"id":"582df522-4cb0-4bb6-8d1a-ce8821f7b00e","arxiv_id":"2511.14675","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Adjoining formal p-adic logarithms of p and 2pi i to the Fargues-Fontaine period ring B makes its Galois cohomology vanish in degrees >=1.","lead":"The authors show that adjoining formal p-adic logarithms of p and 2pi i to the period ring B kills its Galois cohomology in positive degrees. This enables new C_dR and C_st conjectures for compact-support cohomology of p-adic analytic varieties.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.8 rests on an unproved 'descente presque étale' assertion that H^i(H_K,t^jB)=0 for i≥1; this is the load-bearing input for Theorem 2.4 and hence for Conjecture 3.1.","rationale":"The paper's main theorem is a vanishing statement, and the upstream ingredients (Prop 1.2, Prop 2.2, Lemmas 2.1, 2.6, 2.7) are either proved or referenced—except Lemma 2.8. I checked whether Lemma 2.6's product cohomology is problematic; for j≥0 it is fine because H^1(G_K,∏C(-j-1))=0 by Proposition 1.1. I also considered the conjectural avatar B[log ptilde] versus B_st, but that is deliberately conjectural and not needed for the vanishing theorem itself. Thus the unproved almost étale descent is the single load-bearing concern. The reader's assessment already identifies this; my read does not change the verdict.","tokens_in":7169,"tokens_out":13522,"duration_ms":131994,"concrete_test":"Prove Lemma 2.8 in the minimal case that actually tests the descent: show H^1(H_K,tB)=0 by finding b∈tB with (σ−1)b=Kum_p(σ)t for all σ∈H_K, or cite a precise theorem (with all hypotheses checked for B_{[r,s]}) giving H^i(H_K,B_{[r,s]})=0 for i≥1 and justify the passage to B. If no such proof or reference exists, the vanishing theorem and Conjecture 3.1 are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.8 asserts H^i(G_K,t^jB)=0 for all i≥2 and all j∈Z. Its proof is a single sentence: 'par descente presque étale, on prouve que H^i(H_K,t^jB)=0 pour tout i≥1 et tout j', then cd(Γ_K)≤1. No statement of the descent theorem, no verification of its hypotheses, and no reference are supplied. This matters structurally: Proposition 2.9 needs H^2(G_K,tB)=0 to get the surjection onto H^1(G_K,∏C), and Lemma 2.11 needs H^i(G_K,Λ)=0 for i≥2 to run the degree induction that yields H^i(G_K,Λ[log ptilde,logt])=0 for i≥2. Without Lemma 2.8, Theorem 2.4 has no proof, and the compact-support C_st formulation in Conjecture 3.1 loses its stated justification. The descent assertion is not cosmetic: it involves a non-trivial vanishing for H_K acting on the whole Fargues–Fontaine ring, plus a limit from the Banach rings B_{[r,s]} to B; neither step is documented.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Galois cohomology of the period ring B of analytic functions on the Fargues–Fontaine curve (and its localization B[1/t]) and of the de Rham period rings B_dR^+, B_dR. Its main theorem, Théorème 2.4, states that after adjoining formal generators log \\tilde p and log t, the positive-degree continuous Galois cohomology vanishes: H^i(G_K, Λ[log \\tilde p, log t]) = 0 for i ≥ 1, where Λ = B or B[1/t]. The proof proceeds by computing H^1(G_K, B) via a Kummer class and a cyclotomic log class (Proposition 2.9) and then using a polynomial-degree induction (Lemma 2.11) to show that both classes are killed by the adjoined logarithms. The analogous statement for B_dR[log t] is proved in Théorème 1.4. On this basis, the paper formulates Conjecture 3.1, a compact-support analogue of the C_dR and C_st conjectures, using a derived Hom and the rings B_dR[log t] and B[log \\tilde p, log t, 1/t].","tokens_in":7530,"tokens_out":3884,"duration_ms":42900,"significance":"If the main theorem is correct, it removes a well-known obstruction to formulating compact-support p-adic comparison conjectures: the parasitic positive-degree Galois cohomology of period rings. The proposed C_dR/C_st formulation for compact support is natural and is shown to reduce to known statements in the proper case. The paper is concise and the central strategy is transparent: Tate's theorem controls the cohomology of C, and the two logarithms are chosen precisely to eliminate the two H^1 classes. A notable strength is that the vanishing is not assumed; it is derived from Tate's theorem and previously established period-ring results. However, the proof relies on at least one substantial external input—Lemma 2.8's almost étale descent vanishing—that is asserted without proof or reference, and several other cited results are used without verifying their hypotheses. These gaps are load-bearing for the main theorem and for the conjectures that depend on it.","major_comments":[{"comment":"This lemma asserts H^i(G_K, t^j B)=0 for all j and all i≥2. The proof is one sentence: by 'descente presque étale' one proves H^i(H_K, t^jB)=0 for all i≥1, and then cd(Γ_K)≤1. No statement of the descent theorem, no verification of its hypotheses (e.g., the relevant Banach/perfectoid properties of the H_K-action on B, or the limit from B_{[r,s]} to B) and no reference are supplied. This is not cosmetic: Proposition 2.9 needs H^2(G_K,tB)=0 to obtain the surjection onto H^1(G_K,∏ C), and Lemma 2.11 uses H^i(G_K,Λ)=0 for i≥2 to run the induction that yields the main vanishing. Without Lemma 2.8, Theorem 2.4 and hence the justification of Conjecture 3.1 collapse. A complete proof or a precise reference with all hypotheses checked must be supplied.","section":"§2.2, Lemma 2.8"},{"comment":"The proof of Lemma 2.7—that the image of H^1(G_K,tB) in H^1(G_K,B) is one-dimensional generated by the Kummer class of log \\tilde p—relies on [2, prop.10.7], applied to φ^n(c_σ). It is not explained why φ^n(c_σ) satisfies the hypotheses of that proposition, in particular condition (ii) concerning values in B_dR^+. Since Lemma 2.7 is used to describe H^1(G_K,B) in Proposition 2.9 and hence the logarithms that must be adjoined, the missing verification is load-bearing. The authors should either spell out the application of [2, prop.10.7] or give a direct proof.","section":"§2.2, Lemma 2.7"},{"comment":"The injectivity of the GK-equivariant map K⊗_F B[log \\tilde p] → B_dR^+ is cited to [4, prop.10.3.15] without checking that the hypotheses of that proposition are satisfied in the present setting. This injectivity is used in Corollary 2.3(ii) to identify H^0(G_K, B[1/t, logt, log\\tilde p]) = F, which is part of the i=0 case of Theorem 2.4. The authors should indicate how [4, prop.10.3.15] applies, or provide a self-contained argument.","section":"§2.1, Proposition 2.2"},{"comment":"The proof of Théorème 1.4 is only the sentence 'La preuve de la prop.1.2 peut alors s'adapter'. Since this theorem is also needed for the C_dR part of Conjecture 3.1, and the filtration by t^j B_dR^+/t^{j+1}B_dR^+ ≅ C(j) does not immediately behave like the polynomial filtration in Proposition 1.2 (one must handle the inverse/direct limit and the fact that B_dR is not a direct limit of these graded pieces in an obvious way), the reader deserves at least an outline of the adaptation. This is a minor issue if the folklore statement is indeed standard, but it should be made precise.","section":"§1.3, Théorème 1.4"}],"minor_comments":[{"comment":"In the proof of the H^0 statement, the argument 'x:= exp((p−1)p^N a_k)' requires that a_k be sufficiently integral for exp to converge in C; this should be stated explicitly. Also, the notation H0(G_K, C(−j)) on line 3 of p. 3 should be H^0.","section":"§1.2, proof of Prop. 1.2"},{"comment":"The map p^♭ and the element \\tilde p = [p^♭] are used without recalling that ♭ is the tilt of C with respect to p; readers unfamiliar with the Fargues–Fontaine formalism may need a reference or one line of explanation.","section":"§2.1, notation"},{"comment":"The equality H^i(G_K,B[1/t]) = lim_{\\to j} H^i(G_K,t^{-j}B) is stated as immediate. Since these are continuous cohomology groups of a compact group, this is a standard commutation with direct limits, but a brief justification (e.g., via the bar resolution or the fact that G_K is compact) would improve readability.","section":"§2.1, Lemma 2.6"},{"comment":"The reference list, while appropriate, does not include a source for the 'folklore' B_dR^+ statement discussed in the Introduction. If the authors do not wish to prove it in detail, they should at least cite a location where it appears.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a real and recognized obstruction to compact-support p-adic comparison conjectures, and the proposed solution is conceptually attractive. The main concern is not circularity—the vanishing is derived from Tate's theorem and prior period-ring inputs—but incompleteness: the proof of the central Lemma 2.8 is a one-sentence appeal to 'almost étale descent' without a theorem statement, hypothesis check, or reference. This is structurally load-bearing for both Theorem 2.4 and Conjecture 3.1. If the authors can provide a complete proof or a precise reference for Lemma 2.8, and clarify the applications of [2, prop.10.7] and [4, prop.10.3.15], the paper would likely merit acceptance. As it stands, the central claim is not yet established in the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe thing to know: this is a real result. Theorem 2.4, showing H^i(G_K, B[log p-tilde, log t]) = 0 for i≥1 and F in degree 0, is genuinely new, and the compact-support C_dR/C_st formulation in Conjecture 3.1 is a natural but nontrivial extension. The reader's conditional verdict is about right.\n\nWhat's new is the vanishing for B itself. The B_dR[log t] analogue is folklore, but B is the Fargues–Fontaine analytic ring, and adjoining both p-adic logs really does kill the positive-degree cohomology. The strategy is coherent: Tate's description of H^1(G_K,C), the Kummer cocycle, and the two log classes kill the two surviving classes. Lemma 2.6 reducing B[1/t] to B is fine, and Lemma 2.7's computation of the image of H^1(G_K,tB) is plausible, though it leans on a cited proposition.\n\nThe soft spot is Lemma 2.8. Its proof is one sentence: 'par descente presque étale, on prouve que H^i(H_K,t^jB)=0 pour tout i≥1 et tout j', followed by cohomological dimension of Γ_K. No statement of the descent theorem, no reference, no verification of the hypotheses. This is not cosmetic. Lemma 2.8 provides the H^2 vanishing needed in Proposition 2.9 and the i≥2 vanishing used in Lemma 2.11's induction. Without it, Theorem 2.4 has a hole. I don't think the claim is false—it smells like a standard almost étale descent result in this setting—but a referee cannot verify it from the text. The passage from the Banach rings B_{[r,s]} to the inverse limit B is also undocumented here.\n\nA smaller soft spot: Remark 3.2(iv) asserts that replacing B_st^+ by B[log p-tilde] leaves Conjecture 0.1 unchanged, but no proof is given. The conjecture itself is motivated well, and the explanation of why a naive RHom fails is sensible.\n\nNo circularity, no fitted equations. The paper is honest about what is folklore and flags its own limits—for instance, it explains why the same trick does not work for B_cris^+. The value is real: a clean way to kill the Ext^1_G_K terms for de Rham representations, and a concrete conjecture for compact support cohomology.\n\nThis deserves a serious referee. The gap in Lemma 2.8 is fixable, but it needs to be closed before the conjecture is fully supported. I'd cite it once I'm confident that descent step holds up.\n\nRecommendation: send to peer review. The main theorem is significant and the proof strategy is sound; the missing details are a repair, not a wreck.","headline":"A short, inventive note that kills Galois cohomology with two logs and sets up a compact-support C_dR/C_st conjecture; the main theorem is believable but rests on an incompletely documented almost étale descent.","tokens_in":7984,"tokens_out":2413,"would_cite":true,"duration_ms":25370,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F30","14G22","11S25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Adjoining p-adic logarithms of p and 2πi to the ring B on the Fargues-Fontaine curve kills its Galois cohomology in positive degrees, making a compact-support C_dR/C_st conjecture possible.","keywords":["p-adic Hodge theory","Fargues-Fontaine curve","Galois cohomology","period rings","C_dR conjecture","C_st conjecture","compact support cohomology","p-adic logarithms"],"falsifier":"Compute H^2(G_K,t^jB) for a concrete case such as K=Q_p and j=0; a single nonzero class would contradict Lemma 2.8 and therefore Theorem 2.4. More broadly, constructing a nonzero 2-cocycle on G_K with values in B whose class survives after adjoining log p̃ and log t would falsify the paper's central claim.","tokens_in":7105,"feed_emoji":"","tokens_out":12377,"duration_ms":106245,"temperature":0.7,"pith_summary":"This paper proves a vanishing theorem that removes the main obstruction to formulating compact-support versions of the p-adic comparison conjectures. Let B be the ring of analytic functions on the Fargues-Fontaine curve; the paper shows that if one adjoins a p-adic Kummer logarithm of p (log p̃) and a formal logarithm of the p-adic 2πi (log t), the Galois cohomology of B[log p̃, log t] vanishes in degrees ≥1, with H^0 equal to the base field F. The same holds for B[1/t], and for B_dR adjoining log t alone is enough. The reason this matters is that the original C_dR and C_st conjectures work for proper varieties but fail for compact support cohomology because the period rings have nonzero Galois cohomology in degree 1, producing parasitic contributions. Using the new vanishing results, the paper states Conjecture 3.1, a compact-support analogue of C_dR and C_st based on a derived Hom and on the enlarged period ring B[log p̃, log t, 1/t].","feed_headline":"Log p and log 2πi kill Galois cohomology on Fargues-Fontaine","feed_subtitle":"A new vanishing theorem for the period ring B makes compact-support C_dR and C_st conjectures possible.","key_machinery":"The mechanism is the adjunction of two logarithms. log t is a formal transcendental element with Galois action σ(log t) = log t + log χ_cycl(σ), so it turns the classes K·log χ_cycl into coboundaries. log p̃ is a transcendental element over B satisfying φ(log p̃) = p log p̃ and σ(log p̃) = log p̃ + Kum_p(σ)t, where Kum_p is the Kummer cocycle of p; it kills the F·Kum_p class inside H^1(G_K,B). The proof then uses the filtration by total polynomial degree in log p̃ and log t and the quotient descriptions of B/tB to show by induction that the higher cohomology vanishes and the first-cohomology transition maps are zero. The special quotient structure B/tB ≅ ∏ C is what makes B work, and the pap","core_discovery":"The central claim is Theorem 2.4: for Λ = B or B[1/t], one has H^i(G_K, Λ[log p̃, log t]) = F for i=0 and 0 for all i≥1. The proof reduces the cohomology of B to the classical cohomology of the completed algebraic closure C via the exact quotient description B/tB ≅ ∏_{n∈Z} C, identifies H^1(G_K,B) as generated by the Kummer class F·Kum_p together with product-of-K classes times log χ_cycl, and then shows by induction on polynomial degree in the two logarithms that all positive-degree classes are killed. The analogous result for B_dR (Theorem 1.4) requires only log t. These vanishings are exactly what is needed for Conjecture 3.1, which replaces the ordinary Hom in the C_dR/C_st isomorphisms","pith_inferences":["A natural test of Conjecture 3.1 would be to compute both sides for simple non-proper examples, such as the affine line or a punctured curve, where compact-support cohomology and the period-ring Ext groups are explicitly computable; the affine-curve calculation mentioned in the paper is exactly the sort of check that would support or force a modification of the conjecture.","If the unproved almost étale descent assertion behind Lemma 2.8 fails, the compact-support conjecture as stated would need to be reformulated; supplying a proof or a counterexample for that single vanishing statement is the fastest way to test the paper's main conclusion.","The observation that B^+_cris and B^+_rig cannot be fixed by finitely many adjunctions suggests that the compact-support comparison is genuinely tied to the Fargues-Fontaine curve, possibly pointing to a family of period rings parameterized by intervals of the curve for which analogous vanishing and comparison statements could hold."],"forward_implications":["The compact-support C_dR conjecture can now be stated with B_dR[log t]; the adjoined log t trivializes the Ext^1 class that would otherwise double-count de Rham cohomology.","The compact-support C_st conjecture must use the Fargues-Fontaine ring B[log p̃, log t, 1/t] rather than B_st[log t], because H^1(G_K,B_st[log t]) does not vanish; with B[log p̃] the vanishing holds.","When the variety is proper, the new formulation recovers the original Hom-based conjectures, so the traditional C_dR/C_st statements are a special case.","Because H^i(G_K,B) → H^i(G_K,B[1/t]) is an isomorphism (Lemma 2.6), inverting t causes no loss of cohomological information, so the vanishing is stable under localization.","The same logarithmic construction does not work for the more classical rings B^+_cris or B^+_rig; the special quotient structure of B/tB is essential.","A single nonzero class in H^2(G_K,t^jB) would contradict Lemma 2.8 and therefore Theorem 2.4; computing this group for a concrete case such as K=Q_p and j=0 would settle the paper's central claim."],"fun_headline_variants":["Two logs kill Galois cohomology in all positive degrees","Compact-support C_st conjecture enabled by log vanishings","Adding log p and log 2πi extinguishes Galois cohomology","Vanishing theorem for B makes C_dR and C_st conjectures possible","Logarithms annihilate positive-degree Galois cohomology"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The main vanishing theorem depends on Lemma 2.8, which asserts—with no proof or reference, via a one-sentence appeal to 'almost étale descent'—that H^i(G_K,t^jB)=0 for all i≥2; if that assertion fails, Theorem 2.4 and Conjecture 3.1 collapse.","fun_headline_variants_meta":{"raw":{"variants":["Two logs kill Galois cohomology in all positive degrees","Compact-support C_st conjecture enabled by log vanishings","Adding log p and log 2πi extinguishes Galois cohomology","Vanishing theorem for B makes C_dR and C_st conjectures possible","Logarithms annihilate positive-degree Galois cohomology"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000241,"raw_usage":{"total_tokens":1329,"prompt_tokens":685,"completion_tokens":644,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":429,"completion_tokens_details":{"reasoning_tokens":553}},"tokens_in":429,"tokens_out":644,"duration_ms":6634,"temperature":1.0,"reasoning_tokens":553,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T21:32:52.579159+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute H^2(G_K,t^jB) for a concrete case such as K=Q_p and j=0; a single nonzero class would contradict Lemma 2.8 and therefore Theorem 2.4. More broadly, constructing a nonzero 2-cocycle on G_K with values in B whose class survives after adjoining log p̃ and log t would falsify the paper's central claim.","supporting_citations":[],"review_version":1}