{"id":"4494b012-83a2-4ac7-8515-2a87f0246311","arxiv_id":"2509.04954","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Prismatic cohomology of a locally finite free prismatic crystal is canonically isomorphic to A_inf-cohomology of the associated relative Breuil-Kisin-Fargues module, for smooth p-adic formal schemes over the ring of integers of a perfectoid field.","lead":"The paper proves that prismatic cohomology and A_inf-cohomology agree for smooth p-adic formal schemes when the coefficient system is a locally finite free prismatic crystal, extending the known trivial-coefficient comparison. This gives researchers a common language for integral p-adic cohomology with coefficients and opens the path to syntomic complexes and nearby cycles with coefficients.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Global passage from framed affine charts to arbitrary X depends on the non-cartesian simplicial topos direct image of §11 and on [17, Thm 4.26]; a coherence failure there would invalidate Theorem 0.1 even if the local comparison is correct.","rationale":"The reader's verdict is CONDITIONAL, and the weakest assumption identified there includes the non-cartesian descent step. My independent reading agrees that this is the single most load-bearing point. The local computations in §§8–9 are long but contain no identifiable contradiction; the black-box reliance on [17] is a verification gap rather than a demonstrated error. The non-cartesian simplicial direct image is structurally necessary for the global theorem and is admitted by the author to be delicate; if it fails, Theorem 0.1 does not follow even from a correct local comparison. The proposed test isolates this step in a concrete Čeck-nerve setting where the required coherence identities can be checked directly. Since the concern confirms rather than changes the reader's conditional verdict, no adjustment is needed.","tokens_in":84406,"tokens_out":14853,"duration_ms":171287,"concrete_test":"Request the proof of the key §11 lemma (or of [17, Thm 15.5]) in the minimal case of a Zariski Čech nerve of a two-affine cover of a non-affine X (e.g., P^1 over O), with a nontrivial rank-one crystal F. Explicitly construct the non-cartesian direct image functor of §11 on the cosimplicial sheaves, verify that the resulting simple complex is independent of the K-injective resolution, and check that the two composites along the two face maps from level 2 to level 0 induce the same morphism in D^+(X_Zar,A_inf) after applying (8.66). If the square is only commutative up to a non-canonical isomorphism, formula (4.32) is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 0.1 reduces the global case to the local framed comparison (Thm 9.1) via cohomological descent, using Theorem 4.26 and formula (4.32) imported from the companion paper [17], together with the existence of admissible framed embedding systems (Remark 4.25). The author explicitly states in the Introduction that the direct image functor between the simplicial topos components used in this descent is not cartesian and is treated in §11. The load-bearing point is the coherence of the derived functor of this non-cartesian simplicial morphism: one needs Rν_{X_•,t_•,*} to be a well-defined functor on the derived category of the simplicial topos whose simple complex is independent of the chosen resolution, and one needs the comparison maps (8.60)–(8.67) to satisfy the simplicial identities of the Čech nerve. If the §11 construction only provides a lax functor whose coherence data do not match the face/degeneracy maps, then the simple complex in (4.32) is not canonical and the global isomorphism does not follow from the local theorem. Since §11 is not included in the available text and Theorems 4.13/4.20/4.26 are quoted from [17], this is the least secure link in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a comparison theorem (Theorem 0.1 / Theorem 10.1) between prismatic cohomology with locally finite free crystal coefficients and A_inf-cohomology with relative Breuil--Kisin--Fargues module coefficients, for quasi-compact separated smooth p-adic formal schemes over the ring of integers of a perfectoid field containing all p-power roots of unity. The proof proceeds by a local framed comparison via q-Higgs complexes, then a global cohomological-descent argument. The paper also establishes functoriality, compatibility with Frobenius pullback, and compatibility with tensor products.","tokens_in":84581,"tokens_out":4433,"duration_ms":51492,"significance":"If the proof is complete, this is a substantial result: it gives a canonical integral comparison of two cohomology theories with arbitrary locally finite free coefficients, extending Bhatt--Scholze's constant-coefficient comparison and making precise the role of relative Breuil--Kisin--Fargues modules. The local comparison is explicit and the compatibility statements are valuable. A notable strength is that the construction of the comparison map is given in detail in the framed affine case, with an explicit Koszul/q-Higgs description. However, the global theorem depends crucially on results quoted from the author's companion work [17] and on a non-cartesian simplicial descent formalism that is not visible in the available text; without those, the central claim is only conditional.","major_comments":[{"comment":"The local and global descriptions of prismatic cohomology in terms of q-Higgs modules and cohomological descent are quoted from the companion paper [17] without proof. Since these are the structural input for the entire comparison—Theorem 4.26 gives the simplicial version and (4.32) is the descent formula used in the global argument—the paper's main theorem is conditional on the validity of [17]. The author should either include full proofs of these theorems, or state clearly and precisely which results are imported and provide enough detail for the referee to verify their hypotheses are satisfied in the present setting. As it stands, this is a load-bearing gap in the manuscript.","section":"§4, Theorems 4.13, 4.20, 4.26; Remark 4.25; formula (4.32)"},{"comment":"The global passage from the local framed case to arbitrary X uses a direct image functor between simplicial toposes that the author explicitly says is not cartesian and is treated in §11. In the available text, §11 is not included; only its title appears in the table of contents. The canonicality of the simple complex in (4.32) and the coherence of the non-cartesian simplicial direct image Rν_{X_•,t_•,*} are therefore not verifiable from the manuscript. Unless §11 is supplied and shown to satisfy the simplicial identities needed for the Cech-descent comparison, the global isomorphism does not follow from the local Theorem 9.1.","section":"Introduction; §11; Theorem 4.26; (4.32)"},{"comment":"The main theorem is stated as Theorem 0.1 in the Introduction and referenced as Theorem 10.1, but the text of §10 is not present in the submitted version. The global comparison argument, including the role of admissible framed embedding systems and the descent along the simplicial resolution, cannot be assessed. The manuscript should include the complete Section 10, not merely its statement.","section":"Theorem 10.1 / §10"}],"minor_comments":[{"comment":"The notation 'A inf' and 'A_inf' is used inconsistently; it should be typeset uniformly as A_inf. Similarly, 'pro´ etale' appears with a corrupted accent in several places.","section":"Throughout"},{"comment":"The numbering of the main theorem in the Introduction and in Section 10 should be aligned, and the cross-reference should be explicit when the theorem is first stated.","section":"Introduction, Theorem 0.1"},{"comment":"The diagram (8.26) has several unlabeled arrows and the middle vertical map is not explicitly named; adding names would improve readability.","section":"§8, diagram (8.26)"},{"comment":"The phrase 'certain numbers of copies' is vague. Even if the count is irrelevant for the proof, stating the exact multiplicities or at least a formula would make the argument easier to follow.","section":"§9, Lemma 9.16"}],"recommendation":"major_revision","confidential_remarks":"The paper's main theorem is largely conditional on the author's companion work [17], which is cited as a black box for the q-Higgs description and descent. If [17] is not yet publicly available or not yet accepted for publication, the editor should consider whether the dependence should be made explicit in the published version. The absence of §10 and §11 from the available text is a serious obstacle to verification, although the local framed comparison appears carefully written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new result is the coefficient version of the prismatic / A_inf comparison: for a smooth p-adic formal scheme over O_C, a locally finite free prismatic crystal F maps to its relative BKF module M, and Ru_{X/A_inf,*}F is canonically isomorphic to AOmega_X(M). That genuinely extends Bhatt-Scholze and Gaisin-Koshikawa from constant coefficients to locally finite free coefficients. The paper also proves compatibility with pullback, Frobenius, and tensor products; those are not afterthoughts and require real work. The strategy is sound: describe both sides through q-Higgs modules locally, build an explicit comparison morphism via the pro-etale-to-Gamma_Lambda-sheaf morphism, then descend. The local construction is explicit and the writing is careful.\n\nThe soft spots are structural. Theorems 4.13, 4.20, and 4.26 are quoted from the companion [17], and they are load-bearing. Without the q-Higgs description and the descent formula (4.32), the global theorem does not follow from the local one. The author is upfront about this, but an independent referee needs [17] in hand. The same is true for §11: the non-cartesian simplicial direct image is explicitly flagged, but the coherence data needed to make the simple complex in (4.32) canonical are not visible in the available text. That is the least secure link, and the stress-test note correctly identifies it. I did not find a concrete error in §§8-9, though the computations are long and I could not verify every line at referee speed.\n\nOne small thing: the reader's report lists nu_{X,t} as an invented entity. It is not invented; §7 constructs it by evaluating pro-etale sheaves on finite etale covers obtained by adjoining p-power roots of the coordinates. That concern does not land.\n\nIf [17] is solid and the §11 construction checks out, the main theorem is almost certainly correct. The paper deserves a serious referee, but the referee must have access to [17] and the missing part of §11. For a standalone arXiv submission, I would ask the author to state exactly which results from [17] are needed and to include enough of §11 to verify the coherence of the descent. This should not be desk-rejected.","headline":"Serious and credible extension of the prismatic / A_inf comparison to locally finite free coefficients; main risk is the paper's reliance on the companion [17] and the non-cartesian simplicial descent in §11, not the local comparison.","tokens_in":85289,"tokens_out":2833,"would_cite":true,"duration_ms":33819,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F30","14F20","14G22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Theorem 0.1: for a smooth p-adic formal scheme over a perfectoid integer ring, the prismatic cohomology of a locally finite free prismatic crystal is canonically isomorphic to the A_inf-cohomology of the corresponding relative Breuil-Kisin-","keywords":["prismatic cohomology","A_inf-cohomology","Breuil-Kisin-Fargues modules","q-Higgs modules","perfectoid fields","p-adic formal schemes","cohomological descent","prismatic crystals"],"falsifier":"Work out both sides of (0.2) for a genuinely non-constant crystal on a smooth affine formal scheme, e.g., a rank-one crystal on the formal affine line with a specified q-Higgs field: write the q-Higgs complex qΩ•(F_D), form the η_μ-truncated proétale complex K•_Λ(ι^*ν^∞_*M), and check that the comparison map is an isomorphism on every cohomology sheaf; a single nonzero class in a kernel or cokernel annihilated by Ker(A_inf → W(k)) on one side but not the other would refute the local case, and hence the global theorem to which it reduces.","tokens_in":84092,"feed_emoji":"🩼","tokens_out":8680,"duration_ms":82179,"temperature":0.7,"pith_summary":"The paper establishes that two integral p-adic cohomology theories with coefficients agree: for a smooth p-adic formal scheme over the ring of integers of a perfectoid field of mixed characteristic (0,p), the prismatic cohomology of a locally finite free prismatic crystal is canonically isomorphic to the A_inf-cohomology of the relative Breuil-Kisin-Fargues module attached to that same crystal. The comparison is functorial in the crystal and compatible with inverse images, Frobenius scalar extensions, and tensor products, so it identifies the two coefficient theories as objects on the Zariski site of the formal scheme. The proof gives both sides a common local description in terms of q-Higgs complexes on framed small affine charts and assembles the local isomorphisms into a global one by cohomological descent over Zariski hypercoverings. If the theorem is right, computations, structures, and constructions from either side of the comparison transfer to the other, including a triangle relating prismatic cohomology of the crystal to the étale cohomology of the associated lisse Z_p-sheaf after inverting μ.","feed_headline":"Prismatic and A_inf cohomology with coefficients coincide","feed_subtitle":"For smooth p-adic formal schemes, prismatic crystals and Breuil-Kisin-Fargues modules give one canonical comparison isomorphism.","key_machinery":"The argument is carried by three objects working together. (1) The q-Higgs module formalism: on a framed smooth q-prism (D, t, θ_D), a prismatic crystal corresponds to an integrable connection over twisted derivations θ_{D,i}, called a q-Higgs field; its de Rham complex, the q-Higgs complex qΩ•(F_D), computes the prismatic cohomology of F in the affine case (Theorems 4.13 and 4.20). (2) The relative Breuil-Kisin-Fargues module M_{BKF,X}(F), a locally finite free A_inf-module on the proétale site that is trivial modulo μ, with AΩ_X(M) = Lη_μ Rν_{X,*} M. (3) Cohomological descent: the global description of R u_{X/A_inf,*}F via q-Higgs complexes on a cosimplicial envelope of an admissible frame","core_discovery":"The central claim (Theorem 0.1, stated as Theorem 10.1) is that for a quasi-compact, separated, smooth p-adic formal scheme X over the ring of integers O of a perfectoid field C of mixed characteristic (0,p) containing all p-power roots of unity, and for any locally finite free crystal F on the prismatic site (X/(A_inf,(ξ)))_Δ, there is a canonical isomorphism in D⁺(X_Zar, A_inf), functorial in F: R u_{X/A_inf,*} F ≅ AΩ_X(M_{BKF,X}(F)). The right-hand side is the A_inf-cohomology of the relative Breuil-Kisin-Fargues module M assigned to F, defined as Lη_μ of the derived pushforward of M to the Zariski site. The comparison map is constructed one step at a time: locally on framed small affine","pith_inferences":["The q-Higgs descent description is a stronger, more computable object than either cohomology theory alone; it should make explicit calculations feasible on simplicial charts and may carry the full A_inf-cohomology ring structure once combined with the paper's tensor-product compatibility.","The local-to-global strategy suggests a route to relative versions of the theorem for a smooth morphism X → Y, a direction the paper itself poses as a natural question by analogy with known constant-coefficient relative results.","The machinery for sheaves with action of a profinite group and the Koszul resolutions computing the group cohomology of Γ_Λ = Map(Λ, Z_p) is developed independently of the main comparison and could be reused for neighbouring coefficient theories such as syntomic or log-prismatic cohomology."],"forward_implications":["The two integral coefficient theories — prismatic crystals and relative Breuil-Kisin-Fargues modules — produce canonically identical cohomology, so computations and structures from either side transfer.","For a crystal equipped with a Frobenius structure, inverting μ yields an isomorphism (R u_{X/A_inf,*}F)[1/μ] ≅ Rν_{X,*}(L ⊗ A_inf,X)[1/μ], giving a distinguished triangle relating the étale cohomology of the lisse Z_p-sheaf L to the prismatic cohomology of F (0.5).","When X is proper over O and C is algebraically closed, the comparison yields RΓ((X/(A_inf,(ξ)))_{Δ}, F)[1/μ] ≅ RΓ(X_proét, L) ⊗⁽L_{Z_p} A_inf[1/μ], a form of primitive comparison with coefficients.","The compatibility with pullbacks, Frobenius, and tensor products makes the isomorphism a functorial identification of both theories as sheaves of A_inf-modules on the Zariski site, compatible with the ring structure where applicable."],"supporting_citations":[{"why":"Companion paper supplying the q-Higgs module description of prismatic crystals (Theorems 4.13, 4.20) and cohomological descent over admissible framed embedding systems (Theorem 4.26, Remark 4.25) that the global comparison rests on.","marker":"[17]"},{"why":"Introduced relative Breuil-Kisin-Fargues modules and the A_inf-cohomology functor AΩ_X; the construction of M_{BKF,X}(F) and its local cohomological properties in Section 6 come from here.","marker":"[14]"},{"why":"Constant-coefficient prismatic / q-de Rham comparison, giving the base case of (0.2) for F = O_{X/A_inf}, and standard lemmas on prisms and η-operators used throughout.","marker":"[6]"},{"why":"Foundational source for A_inf-cohomology and the Lη_μ operation appearing in the definition of AΩ_X(M).","marker":"[5]"},{"why":"Perfectoid-space results and the primitive comparison theorem used to derive the proétale/étale corollaries (0.4)–(0.6) and the structural properties of M in Section 6.","marker":"[15]"}],"fun_headline_variants":["Prismatic crystals unlock A_inf cohomology","Prismatic and A_inf cohomology coincide with coefficients","Smooth p-adic schemes: prismatic equals A_inf cohomology","Crystal-to-module bridge: prismatic to A_inf cohomology","Canonical isomorphism: prismatic crystals to A_inf"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The global comparison is assembled from a black-box description of prismatic cohomology by q-Higgs complexes over an admissible framed embedding system, taken from the companion paper: if that descent description, or the treatment of non-cartesian direct images of simplicial topos, fails, the local framed isomorphisms do not glue into Theorem 0.1.","fun_headline_variants_meta":{"raw":{"variants":["Prismatic crystals unlock A_inf cohomology","Prismatic and A_inf cohomology coincide with coefficients","Smooth p-adic schemes: prismatic equals A_inf cohomology","Crystal-to-module bridge: prismatic to A_inf cohomology","Canonical isomorphism: prismatic crystals to A_inf"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1246,"prompt_tokens":718,"completion_tokens":528,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":462,"completion_tokens_details":{"reasoning_tokens":436}},"tokens_in":462,"tokens_out":528,"duration_ms":4693,"temperature":1.0,"reasoning_tokens":436,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T05:45:38.052882+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Work out both sides of (0.2) for a genuinely non-constant crystal on a smooth affine formal scheme, e.g., a rank-one crystal on the formal affine line with a specified q-Higgs field: write the q-Higgs complex qΩ•(F_D), form the η_μ-truncated proétale complex K•_Λ(ι^*ν^∞_*M), and check that the comparison map is an isomorphism on every cohomology sheaf; a single nonzero class in a kernel or cokernel annihilated by Ker(A_inf → W(k)) on one side but not the other would refute the local case, and hence the global theorem to which it reduces.","supporting_citations":[{"cited_title":"Bhatt and P","cited_arxiv_id":null,"evidence_quote":"Constant-coefficient prismatic / q-de Rham comparison, giving the base case of (0.2) for F = O_{X/A_inf}, and standard lemmas on prisms and η-operators used throughout."},{"cited_title":"Bhatt, M","cited_arxiv_id":null,"evidence_quote":"Foundational source for A_inf-cohomology and the Lη_μ operation appearing in the definition of AΩ_X(M)."},{"cited_title":"Scholze,p-adic Hodge theory for rigid-analytic varieties—corrigendum [MR3090230], Forum Math","cited_arxiv_id":null,"evidence_quote":"Perfectoid-space results and the primitive comparison theorem used to derive the proétale/étale corollaries (0.4)–(0.6) and the structural properties of M in Section 6."}],"review_version":1}