{"id":"b536cb5b-fba2-4a9d-858a-c7b2aa5cbe7d","arxiv_id":"2507.08451","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Breuil-Kisin cohomology of analytic log prismatic F-crystals on semistable p-adic formal schemes is canonically isomorphic, after tensoring with A_inf and inverting mu, to the etale cohomology of the corresponding semistable Z_p-local systems.","lead":"This paper proves a comparison theorem: for semistable p-adic formal schemes, the prismatic cohomology of a natural crystal matches the etale cohomology of the associated local system. It extends work of Guo and Reinecke from smooth to semistable varieties, including the non-proper case, which is new even for smooth schemes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 0.4(2) depends on a coefficient version of Scholze's primitive comparison theorem that is cited but not proved; the cited [20, Thm 5.1] must be checked to cover arbitrary finite free Z_p-local systems, not just the constant sheaf.","rationale":"The paper is a serious, detailed proof, and I did not find an internal contradiction in the local q-Higgs and Galois-cohomology computations of Sections 3 and 4. The self-flagged issue in the footnote to Theorem 6.18 is handled by working only with the Čech nerve Rrel,•_S, so it does not appear to invalidate the finiteness result. The main proof is nevertheless built on several external inputs. The reader's weakest_assumption lists both the Du–Liu–Moon–Shimizu equivalence and the limit version of the primitive comparison theorem. I agree with that assessment, but I would single out the coefficient version of [20, Thm 5.1] as the most load-bearing unproved step for the proper étale comparison, because the paper cites it without proving that it applies to an arbitrary T(E). If the cited theorem already covers local systems, the concern is resolved; if not, Theorem 0.4(2) lacks a necessary ingredient. This does not change the appropriate verdict: the submission should remain conditional pending independent verification of the cited inputs, rather than being accepted outright or rejected on internal grounds.","tokens_in":59400,"tokens_out":29737,"duration_ms":365219,"concrete_test":"Locate the exact statement of [20, Theorem 5.1] and verify whether it applies to arbitrary finite free Z_p-local systems on proper smooth rigid spaces over C. If it does, the step in Theorem 5.6(2) is justified. If it is constant-coefficient only, supply the missing coefficient argument by tracing the Leray spectral sequence for a nontrivial local system L on a proper smooth curve and checking that the edge map becomes an isomorphism after inverting μ; if the spectral sequence fails to degenerate, the claimed almost isomorphism does not follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proper case of the main theorem is obtained by combining Theorem 5.6(2) with Proposition 7.4. In the proof of Theorem 5.6(2), after reducing to pro-étale cohomology, the paper invokes 'a limit version of Scholze's primitive comparison theorem [20, Theorem 5.1]' to pass from H^i(X_η,proét, T(E) ⊗_{Z_p} A_inf,X_η) to H^i(X_η,ét, T(E)) ⊗_{Z_p} A_inf, almost after inverting μ. No proof or further reference is given for this local-system version. If [20, Thm 5.1] is stated only for constant coefficients, the step from Z_p to an arbitrary finite free Z_p-local system T(E) requires an additional argument: one must justify the limit over p^n, handle the A_inf-twist, and control the relevant spectral sequence or descent. The paper does not supply such an argument. Since this is precisely the ingredient that produces the étale side of Theorem 0.4(2), the strongest form of the main theorem is conditional on this external input. The reader's verdict flagged external inputs generally; the present concern isolates the coefficient-version issue, which is the most load-bearing unproved step in the passage from local prismatic computations to the final étale comparison. If [20, Thm 5.1] or a standard consequence does cover arbitrary finite free Z_p-local systems on proper smooth rigid spaces over C, this objection disappears and the proof's internal structure appears coherent.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a semistable prismatic–étale comparison theorem. For a separated semistable p-adic formal scheme X over Spf(O_K) and an analytic prismatic F-crystal E, it establishes a canonical isomorphism between the p-adically completed Breuil–Kisin cohomology of E base-changed to A_inf (after inverting mu) and the pro-étale cohomology of the associated semistable Z_p-local system T(E); in the proper case it identifies this with étale cohomology of T(E) after inverting mu. The proof proceeds through local q-Higgs descriptions (Section 3), a Galois-cohomology and décalage bridge (Section 4), a comparison over O_C using perfect prismatic sites and diamonds (Section 5), the construction of the canonical extension j_* and finiteness of Breuil–Kisin cohomology (Section 6), and base change from O_K to O_C (Section 7). The argument is written in full, with explicit framings, Čech–Alexander complexes, and descent arguments.","tokens_in":59683,"tokens_out":19421,"duration_ms":220603,"significance":"If the cited external inputs are valid, the theorem is a substantial generalization of Guo–Reinecke's prismatic–étale comparison to the semistable case, and it provides a direct proof of the strong étale comparison without invoking Poincaré duality. The manuscript is parameter-free and self-contained in its internal structure: the local computations in Section 3, the Galois-cohomology bridge in Section 4, and the base-change and descent arguments in Sections 5–7 are all given explicitly. The main caveats are two black-box external inputs: the Du–Liu–Moon–Shimizu equivalence [10, Cor. 5.2] used to define the local system T(E), and the cited primitive comparison theorem of Scholze used in Theorem 5.6(2). Neither is proved in this paper, so the strongest form of the main theorem is conditional on these inputs.","major_comments":[{"comment":"The passage from pro-étale to étale cohomology is load-bearing for Theorem 0.4(2) and is the one place where the manuscript is not self-contained: it invokes 'a limit version of Scholze's primitive comparison theorem [20, Theorem 5.1]' without stating the version used. If [20, Thm. 5.1] is stated only for the constant sheaf, an additional argument is required for an arbitrary finite free Z_p-local system T(E): one must justify the limit over p^n, handle the A_inf-twist, and control the relevant spectral sequence or descent. Please either quote the exact theorem from [20] in the form needed here or supply the missing reduction. If [20, Thm. 5.1] does cover lisse Z_p-sheaves, this comment is resolved by adding the precise statement and a short indication of why the cited theorem applies to T(E).","section":"§5.6, proof of Theorem 5.6(2)"},{"comment":"Theorem 0.4 and Theorem 5.6(2) depend essentially on the Du–Liu–Moon–Shimizu equivalence T: Vectan(X_Delta, O_Delta)^(phi=1) -> Loc_st^{Z_p}(X_eta,et) and on the identification of the essential image with semistable étale Z_p-local systems. The manuscript cites [10, Cor. 5.2] and [10, Prop. 3.21] but does not state the precise hypotheses under which these results apply to separated (not necessarily proper) semistable formal schemes over a complete discrete valuation ring. Since the main theorem inherits the full validity of this external equivalence, please state explicitly the exact input from [10] and, if [10] is a preprint, the version used.","section":"§7.6 and §0.2"}],"minor_comments":[{"comment":"The footnote 'I don’t know whether j_*(E)/E(u)j_*(E) ≃ j_*E' is an explicit limitation statement. The proof as written uses only the Čech nerve R^{rel,•}_S, where the needed isomorphisms hold, so the argument appears to go through; however, the reader is left to verify that the unknown sheaf-level assertion is not used elsewhere. Please add a clarifying remark after the footnote explaining that the subsequent finiteness proof requires only the Čech-nerve version.","section":"§6, footnote in proof of Theorem 6.18"},{"comment":"The manuscript contains numerous typographical errors that should be corrected in revision, including 'caononical', 'moprhism', 'obejct', 'funtor', 'equippd', 'Lebnitz', 'autormphism', and 'ismorphism'.","section":"Throughout"},{"comment":"The set Xi_p is introduced in (4.3.1) and then Xi^*_p is used in the proof of Theorem 4.6(2) without an explicit definition; please define Xi^*_p immediately after (4.3.1) and state its role in the decomposition of E_infty.","section":"§4.3 and §4.6"},{"comment":"The citation [20, Theorem 5.1] should include a page or theorem number and, ideally, the precise statement of the version used, since the proof of Theorem 5.6(2) relies on a 'limit version' that is not quoted.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is strong and the internal structure appears coherent, but the main theorem is conditional on two external results, both of which are recent preprints in the field. The editor may wish to verify the status of [10] and [20] before acceptance. The primitive comparison step in Theorem 5.6(2) is the most load-bearing unproved input and should be made precise; if the cited theorem already covers local systems, the revision is straightforward."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The semistable prismatic-etale comparison is real and the proof is mostly solid, but the proper case rests on an unproved external input that the paper cites rather than establishes. Tian extends Guo-Reinecke's smooth comparison to semistable formal schemes and adds a non-proper version, Theorem 0.4(1), which is new even in the smooth case. That is genuine progress. The proof strategy is also different from Guo-Reinecke: it goes through the perfect prismatic site and a detailed q-Higgs computation, with local calculations in Section 3, a Galois-cohomology bridge in Section 4, and descent/base-change in Sections 5-7. I did not find internal contradictions or fitting. Self-citations to the author's earlier work are only for auxiliary lemmas; the main weight is on external theorems [10], [16], [24], and [20]. The author also honestly flags the delicate p-adic completeness issue in Section 6 and resolves it with Cech nerves, which is the right way to behave. The soft spot is exactly what the stress-test note isolates. In Theorem 5.6(2), the passage from pro-etale to etale cohomology invokes 'a limit version of Scholze's primitive comparison theorem [20, Theorem 5.1]' for an arbitrary finite free Z_p-local system T(E), not just the constant sheaf. The paper does not state or prove that coefficient version, and [20, Thm 5.1] as written may only handle constant coefficients. If so, one needs an additional argument via limits over p^n, the A_inf twist, or a spectral sequence; none is supplied. This is the load-bearing step for Theorem 0.4(2), so the strongest form of the theorem is conditional. The reader's CONDITIONAL verdict is right. The non-proper statement (1) is less exposed, but it also inherits the DLMS equivalence [10, Cor 5.2] and the same pro-etale to etale mechanism after inverting mu. If [20, Thm 5.1] or a standard consequence does cover finite free local systems on proper smooth rigid spaces, the objection disappears and the paper's internal structure looks coherent. But the paper should say so explicitly. Who is this for? Specialists in p-adic Hodge theory, especially people working on prismatic cohomology and semistable local systems. It deserves a serious referee, not a desk reject. I would send it to peer review with a clear request: verify the coefficient version of [20, Thm 5.1], or ask the author to supply the missing argument. If that step is filled, the theorem is a solid contribution.","headline":"A serious, detailed proof of the semistable prismatic-etale comparison for analytic F-crystals; the main theorem is conditional on a cited coefficient version of Scholze's primitive comparison that the paper does not prove.","tokens_in":747,"tokens_out":843,"would_cite":true,"duration_ms":34381,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F30","14G20","14F20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For semistable p-adic formal schemes, Breuil–Kisin cohomology of analytic prismatic F-crystals is canonically isomorphic, after base change to A_inf and inverting µ, to the étale cohomology of the corresponding semistable Z_p-local system.","keywords":["prismatic cohomology","semistable formal schemes","log prismatic site","Breuil–Kisin cohomology","étale comparison","p-adic Hodge theory","analytic F-crystals","q-Higgs modules"],"falsifier":"A concrete falsifier would be to compute both sides for a specific separated non-proper semistable example—for instance, an affine annulus built from $T_0T_1 = \\pi$—and a rank-one analytic prismatic F-crystal whose local system has nontrivial monodromy: if the canonical map of Theorem 0.4(1) were not an isomorphism after inverting $\\mu$, or the Frobenius and Galois actions disagreed, the theorem would fall. Since the left side is explicitly computable via $q$-de Rham complexes and the right side via étale cohomology of the local system, a discrepancy in the first cohomology group would settle it.","tokens_in":59125,"feed_emoji":"🧮","tokens_out":10741,"duration_ms":105458,"temperature":0.7,"pith_summary":"The paper establishes a semistable analogue of the prismatic–étale comparison theorem: for a separated semistable p-adic formal scheme, the Breuil–Kisin cohomology of an analytic prismatic F-crystal becomes, after base change to the period ring $A_{\\mathrm{inf}}$ and inverting the element $\\mu$, canonically isomorphic to the pro-étale cohomology of the semistable étale $\\mathbb{Z}_p$-local system attached to the crystal through the [10] equivalence. For proper schemes the isomorphism is precisely on each cohomology group, étale rather than pro-étale, and equivariant under Frobenius and Galois actions. This generalizes the crystalline prismatic–étale comparison of [12] to the semistable setting and gives a comparison in the non-proper case that is new even for smooth schemes. A reader should care because it pins down the integral cohomology of semistable local systems in terms of prismatic coefficients, a step toward a full prismatic-crystalline bridge for semistable representations.","feed_headline":"Semistable prismatic–étale comparison proved","feed_subtitle":"Breuil–Kisin cohomology now matches étale cohomology of semistable local systems, even for non-proper schemes.","key_machinery":"The load-bearing machinery is the absolute log prismatic site $X_\\Delta$ of the semistable log formal scheme together with its analytic prismatic F-crystals; the Breuil–Kisin log prism $\\mathbb{S} = (S,(E(u)),\\mathbb{N})$, whose Čech–Alexander complex defines Breuil–Kisin cohomology; and a local $q$-Higgs description: for small affine charts over $\\mathrm{Spf}(O_C)$, the cohomology of a complete prismatic crystal is computed by the $q$-de Rham complex of an attached topologically quasi-nilpotent $q$-Higgs module. From that local description the paper derives a comparison with Galois cohomology via the décalage functor $L\\eta_\\mu$, transfers it to the perfect prismatic site through the equivalence of Prop. 1.23, and finishes by passing from pro-étale to étale cohomology using the primitive comparison theorem of [20].","core_discovery":"The central discovery is Theorem 0.4: for a separated semistable p-adic formal scheme $X$ over $\\mathrm{Spf}(O_K)$ and an analytic prismatic F-crystal $E$ on its absolute log prismatic site, there is a canonical, Frobenius- and Galois-equivariant isomorphism $(R\\Gamma_S(X,E) \\otimes^L_S A_{\\mathrm{inf}})^\\wedge[1/\\mu] \\simeq R\\Gamma(X_{C,\\mathrm{proet}}, T(E) \\otimes_{\\mathbb{Z}_p} A_{\\mathrm{inf},X_C})[1/\\mu]$, where the left side is the Breuil–Kisin cohomology of $E$ base-changed to $A_{\\mathrm{inf}}$ and completed, and the right side is the pro-étale cohomology of the semistable local system $T(E)$ obtained from $E$ via the equivalence of [10, Cor. 5.2]. When $X$ is proper over $\\mathrm{Spf}(O_K)$, the completion is unnecessary and one obtains $R\\Gamma_S(X,E) \\otimes^L_S A_{\\mathrm{inf}}[1/\\mu] \\simeq R\\Gamma(X_{C,\\mathrm{et}}, T(E)) \\otimes^L_{\\mathbb{Z}_p} A_{\\mathrm{inf}}[1/\\mu]$, hence an isomorphism on each cohomology group. The proof compares the prismatic side to the perfect prismatic site, computes cohomology locally through $q$-Higgs modules and Galois cohomology, and then invokes the primitive comparison theorem of [20] to land in étale cohomology.","pith_inferences":["A natural next step is to test whether the same local $q$-Higgs technology extends the comparison to arbitrary log smooth p-adic formal schemes, with the semistable divisor replaced by a general snc boundary.","If a conjectural prismatic–crystalline comparison for the attached crystalline F-isocrystal is filled in, the canonical isomorphism of Theorem 0.4 would yield the classical semistable comparison at integral level, and the monodromy filtrations on both sides should match.","A concrete new case to probe is a relative annulus or product of a semistable curve with itself, where explicit $q$-de Rham and explicit étale cohomology can be computed by hand; agreement of the Galois action on the resulting $A_{\\mathrm{inf}}[1/\\mu]$-modules would support the theorem beyond the proper case."],"forward_implications":["For proper semistable $X$, each Breuil–Kisin cohomology group $H^i_S(X,E)$ is a finitely generated $S$-module, and after base change to $A_{\\mathrm{inf}}[1/\\mu]$ it is isomorphic to $H^i(X_{C,\\mathrm{et}},T(E)) \\otimes_{\\mathbb{Z}_p} A_{\\mathrm{inf}}[1/\\mu]$; integral étale cohomology of semistable local systems is thereby determined by prismatic data.","The non-proper statement gives a derived comparison for separated semistable formal schemes, with the left side taken up to $(p,\\mu)$-completion; this is new even when $X$ is smooth and non-proper.","The isomorphism is equivariant under Frobenius and the Galois group of an algebraic closure of $K$ over $K$, so arithmetic actions on the two sides are matched, not just the underlying abelian groups.","Taking $E = O_\\Delta$ recovers, after identification, the $A_{\\mathrm{inf}}$-cohomology descriptions for constant coefficients in the semistable case.","Combined with the equivalence of categories [10, Cor. 5.2], the theorem upgrades that equivalence from an identification of objects to an identification of cohomology theories."],"supporting_citations":[{"why":"Supplies the equivalence between analytic prismatic F-crystals and semistable étale Z_p-local systems that defines T(E) on the right-hand side.","marker":"[10, Cor. 5.2]"},{"why":"The crystalline prismatic–étale comparison theorem this paper generalizes, together with a lemma used in the direct proof.","marker":"[12, Thm. 9.1]"},{"why":"Provides the limit primitive comparison theorem used to pass from pro-étale to étale cohomology after inverting µ.","marker":"[20, Thm. 5.1]"},{"why":"Gives the local q-Higgs/de Rham description of prismatic cohomology that the semistable argument adapts.","marker":"[24, Thm. 11.20]"},{"why":"Establishes the equivalence between perfect log prismatic and non-log perfect prismatic sites, used to transfer to the perfect side.","marker":"[16, Prop. 2.18]"},{"why":"Supplies the almost-isomorphism and décalage facts used to replace perfectoid rings by their values under A_inf.","marker":"[3, Lemma 3.21]"},{"why":"Gives the flatness and prismatic-envelope facts that make Čech–Alexander complexes compute prismatic cohomology.","marker":"[5, Prop. 3.13]"},{"why":"Provides the log prismatic site framework and the equivalence for semistable local systems for constant coefficients.","marker":"[14, Thm. 7.36]"},{"why":"Identifies A_inf-cohomology with log prismatic cohomology for the constant crystal, recovered as a special case.","marker":"[7, Thm. 2.3]"}],"fun_headline_variants":["Semistable prismatic-etale comparison established","Breuil-Kisin cohomology equals etale cohomology semistably","Prismatic-etale comparison covers semistable case","Semistable generalization of prismatic-etale comparison"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem leans on two external results that it cites rather than proves—the equivalence between analytic prismatic F-crystals and semistable étale Z_p-local systems of [10, Cor. 5.2], and the limit version of the primitive comparison theorem of [20, Thm. 5.1]—so the central claim is only as secure as those two inputs for the separated, possibly non-proper semistable schemes considered here.","fun_headline_variants_meta":{"raw":{"variants":["Semistable prismatic-etale comparison established","Breuil-Kisin cohomology equals etale cohomology semistably","Prismatic-etale comparison covers semistable case","Semistable generalization of prismatic-etale comparison"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0006,"raw_usage":{"total_tokens":2879,"prompt_tokens":1095,"completion_tokens":1784,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":711,"completion_tokens_details":{"reasoning_tokens":1716}},"tokens_in":711,"tokens_out":1784,"duration_ms":14750,"temperature":1.0,"reasoning_tokens":1716,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:18:57.559469+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete falsifier would be to compute both sides for a specific separated non-proper semistable example—for instance, an affine annulus built from $T_0T_1 = \\pi$—and a rank-one analytic prismatic F-crystal whose local system has nontrivial monodromy: if the canonical map of Theorem 0.4(1) were not an isomorphism after inverting $\\mu$, or the Frobenius and Galois actions disagreed, the theorem would fall. Since the left side is explicitly computable via $q$-de Rham complexes and the right side via étale cohomology of the local system, a discrepancy in the first cohomology group would settle it.","supporting_citations":[],"review_version":1}