{"id":"de32929a-2077-411c-80be-c7ccdf3d8565","arxiv_id":"2607.08965","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Functorial balanced big Cohen-Macaulay algebra assignments factor through RΓ(Y, O_Y) for every proper birational map Y → Spec R.","lead":"Any sufficiently functorial big Cohen-Macaulay algebra of a reduced equidimensional local ring factors, in the derived category, through the derived global sections of every proper birational model. This turns a property previously known only for absolute integral closures into a formal consequence of weak functoriality.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the functoriality hypothesis as the sole external input and correctly assesses the remainder of the argument as self-contained classical commutative algebra. The two proofs of Lemma 3.2 (Cousin filtration and middle-perverse t-structure) supply independent routes to the same vanishing, and Remark 3.6 gives a third route under completeness. No load-bearing technical flaw is present that would warrant changing the ACCEPT verdict or lowering confidence.","tokens_in":11628,"tokens_out":413,"duration_ms":4752,"concrete_test":"Independently verify that the support estimate dim Supp H^j(K^•) ≤ n-j (display (3.0.1)) holds after localizing at an arbitrary prime of height j, and that the resulting localized diagram still satisfies the hypotheses of Theorem 3.1; if the estimate fails for some equidimensional reduced example, the application of Lemma 3.2 would be blocked.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is conditional on a weakly functorial BCM assignment compatible with surjections from localizations of Rees algebras (Definition 2.5 and the diagram in Theorem 3.5). That hypothesis is known to hold in the settings where BCM algebras exist (via R+, cR+, or André’s constructions) and is stated explicitly; it is not a hidden gap. The reduction from proper birational maps to blow-ups via Chow’s lemma (Corollary 3.7), the support bound (3.0.1), the Sancho-de-Salas comparison (Theorem 3.1), and the two independent proofs of the key vanishing lemma (Lemma 3.2) are classical and carefully written. No internal inconsistency or unstated assumption that would break the factorization appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that if (R, m) is a reduced universally catenary equidimensional Noetherian local ring and R ↦ B_R is a sufficiently weakly functorial balanced big Cohen-Macaulay algebra assignment (compatible at least with surjections from localizations of Rees algebras), then for every proper birational map Y \to Spec R the structure map R \to B_R factors in D(R) through RΓ(Y, O_Y). The argument reduces via Chow’s lemma to the blow-up of an ideal of positive height, compares the fiber of R \to RΓ(Y, O_Y) with the degree-zero part of a Sancho-de-Salas sequence (Theorem 3.1), and then shows that any map from that fiber into a weakly cohomologically Cohen-Macaulay module that vanishes after local cohomology at every prime is already zero in D(R) (Lemma 3.2). Two independent proofs of the vanishing lemma are given (Cousin filtration and middle-perverse t-structure). The result recovers that weakly BCM-regular rings are birational derived splinters and, combined with earlier work, the Briançon–Skoda theorem of Rodríguez-Villalobos–Schwede.","tokens_in":11780,"tokens_out":845,"duration_ms":8897,"significance":"The factorization property was previously known only for the concrete constructions R^{+} (char p) and ĉR^{+} (mixed characteristic). Establishing it for any weakly functorial BCM assignment unifies those results and supplies a formal reason why the property holds whenever such algebras exist. The argument is short, classical (Cousin complexes, Sancho-de-Salas, perverse t-structures), and supplies two independent proofs of the key vanishing statement. The applications to derived splinters and Briançon–Skoda are immediate and clean. The manuscript is therefore a useful structural contribution to the theory of big Cohen-Macaulay algebras.","major_comments":[],"minor_comments":[{"comment":"Definition 2.5 introduces a “weak CM-assignment” while the abstract and introduction speak of “sufficiently functorial” or “weakly functorial” balanced big Cohen-Macaulay algebra assignments. A single sentence equating the two terminologies would avoid any momentary confusion.","section":null},{"comment":"In the proof of Theorem 3.1 the identification [RΓ_{S>0} S]_0 ≅ K• is asserted by reference to Lipman; a one-line reminder that the degree-zero part of the Sancho-de-Salas sequence is precisely the fiber of R \to RΓ(Y, O_Y) would make the comparison self-contained.","section":null},{"comment":"Lemma 2.2 sketches the weakly balanced case; the balanced case is cited to the literature. Since the main theorems only need the weakly balanced/cohomological version, the sketch is sufficient, but a parenthetical remark that the balanced localization statement is classical would be helpful.","section":null},{"comment":"The AI-acknowledgement section is unusually detailed. It is honest and does not affect the mathematics, but the journal may wish to decide whether such a section should appear in the published version or be moved to a supplementary note.","section":null},{"comment":"Typographical: “Sancho de Salas” is sometimes hyphenated and sometimes not; “cR+” versus “ĉR+” appears inconsistently in the introduction. Standardize throughout.","section":null}],"recommendation":"accept","confidential_remarks":"The mathematical content is solid and the result is of genuine interest to the commutative-algebra community. The only non-mathematical novelty is the unusually frank AI-acknowledgement; I leave it to the editor whether that section should be retained, shortened, or relocated. No other concerns about novelty, citation practice, or scope."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"Karl extracts a property that was known only for R+ and its p-adic completion and shows it is formal for any weakly functorial balanced big Cohen-Macaulay assignment. The statement is clean: for reduced universally catenary equidimensional local R, the map R to B_R factors through RGamma(Y, O_Y) for every proper birational Y. That immediately recovers the derived-splinter property for weakly BCM-regular rings and, with earlier work, the Briançon-Skoda results that used the same factorization.\n\nWhat is new is the combination. Theorem 3.1 uses a Sancho-de-Salas comparison to get local-cohomology vanishing of the fiber of R to RGamma(Y, O_Y). Lemma 3.2 then upgrades that vanishing, via either a Cousin filtration (Sharp) or the middle-perverse t-structure (Bhatt’s sketch), to a global vanishing in the derived category once the support-dimension bound (3.0.1) is in hand. The reduction from proper birational maps to blow-ups is the usual Chow-lemma argument. Two independent proofs of the key lemma are written out; the support estimates are careful; the localization statements for weakly balanced BCM modules are standard and correctly applied.\n\nThe only real limitation is the hypothesis itself: you need a weakly functorial assignment that is compatible with surjections from localizations of Rees algebras. That is known in the settings where BCM algebras exist, and the paper states it explicitly rather than hiding it. There is no circularity, no free parameters, and no invented objects. The AI-acknowledgement section is unusually candid but does not affect the mathematics; the final arguments are classical and human-verified.\n\nThis is for people who work with BCM algebras, mixed-characteristic singularities, or derived splinters. It is short, self-contained once Sharp and Sancho-de-Salas are granted, and it organizes several recent theorems as formal consequences. I would send it to a serious referee without hesitation and would cite the main theorem when I need the factorization for a general BCM assignment rather than just for absolute integral closures.","headline":"Clean formal extraction: any weakly functorial BCM assignment factors through derived global sections of every proper birational model.","tokens_in":12376,"tokens_out":526,"would_cite":true,"duration_ms":6469,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D22","13H10","14B05","13A30"],"pacs":[],"model":"grok-4.5","headline":"Any sufficiently functorial big Cohen-Macaulay algebra factors through the derived global sections of every proper birational map.","keywords":["big Cohen-Macaulay algebras","derived splinters","Rees algebras","Cousin complexes","birational morphisms","weak functoriality","local cohomology"],"falsifier":"Exhibit a reduced equidimensional local ring R together with a weakly functorial balanced big Cohen-Macaulay algebra B_R and a blow-up Y \to Spec R for which the composition of the structure map with the natural map R \to RΓ(Y, O_Y) fails to vanish on the fiber in the derived category.","tokens_in":12505,"feed_emoji":"∞","tokens_out":975,"duration_ms":10620,"temperature":0.7,"pith_summary":"The paper shows that maps from a reduced equidimensional Noetherian local ring into a big Cohen-Macaulay algebra always factor, in the derived category, through the derived global sections of any proper birational modification of the spectrum. Big Cohen-Macaulay algebras have long been a tool for proving vanishing and purity statements in commutative algebra; the new result says that once such an algebra is assigned in a weakly functorial way, the factorization through blow-ups (and more generally through proper birational maps) is automatic. The argument first reduces to the Rees algebra of an ideal of positive height, then uses a Sancho-de-Salas comparison to kill the fiber of the structure map after local cohomology, and finally employs a Cousin filtration argument to promote that vanishing to an actual factorization. The payoff is that several known purity and splinter properties of rings that admit big Cohen-Macaulay algebras become formal consequences of the assignment itself rather than of special constructions such as absolute integral closures.","feed_headline":"Big Cohen-Macaulay algebras factor through every blowup","feed_subtitle":"The factorization is formal once the algebra assignment is weakly functorial for Rees surjections","key_machinery":"The combination of a Sancho-de-Salas triangle for the Rees algebra of a positive-height ideal with Sharp’s characterization of weakly balanced big Cohen-Macaulay modules by exactness of the height-filtration Cousin complex; together they convert local-cohomology vanishing into a global factorization.","core_discovery":"For a reduced universally catenary equidimensional Noetherian local ring R, any weakly functorial balanced big Cohen-Macaulay algebra assignment R ↦ B_R, and any proper birational map Y → Spec R, the structure map R \to B_R factors in the derived category as R \to RΓ(Y, O_Y) \to B_R.","pith_inferences":["If the assignment can be upgraded so that the maps are of derived commutative algebras, the factorization would also control higher multiplicative structure on the derived global sections.","The same formal argument may apply to other classes of algebras characterized by vanishing of local cohomology (for instance certain perfectoid or almost Cohen-Macaulay algebras) once weak functoriality for Rees surjections is verified.","Question 3.9 in the paper suggests a natural next test: whether the factorization can be realized by maps of derived rings rather than merely of complexes."],"forward_implications":["Weakly BCM-regular rings are automatically birational derived splinters.","The main purity statements previously proved for absolute integral closures or their p-adic completions become formal consequences of any sufficiently functorial BCM assignment.","The Briançon–Skoda theorem and related integral-closure results for pseudo-rational and Du Bois singularities follow from the same formal factorization once a weakly functorial BCM algebra is known to exist.","The same factorization holds after localization, so the result globalizes to any reduced universally catenary locally equidimensional Noetherian ring that admits a weak CM-assignment."],"fun_headline_variants":["Big CM algebras factor through every proper birational map","Maps to big Cohen-Macaulay algebras factor via blowups","Weakly functorial big CM algebras factor through all blowups","Structure maps to big CM algebras factor through RΓ of blowups","Big Cohen-Macaulay algebra assignments factor through birational maps"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The existence of a balanced big Cohen-Macaulay algebra assignment that is weakly functorial at least for surjections coming from localizations of Rees algebras; without that commutative diagram the comparison has nothing to map into.","fun_headline_variants_meta":{"raw":{"variants":["Big CM algebras factor through every proper birational map","Maps to big Cohen-Macaulay algebras factor via blowups","Weakly functorial big CM algebras factor through all blowups","Structure maps to big CM algebras factor through RΓ of blowups","Big Cohen-Macaulay algebra assignments factor through birational maps"]},"model":"grok-4.5","effort":"low","cost_usd":0.009594,"raw_usage":{"total_tokens":2079,"prompt_tokens":624,"num_sources_used":0,"completion_tokens":89,"cost_in_usd_ticks":95940000,"prompt_tokens_details":{"text_tokens":624,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1366,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":624,"tokens_out":89,"duration_ms":20823,"temperature":1.0,"reasoning_tokens":1366,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T05:26:13.609844+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a reduced equidimensional local ring R together with a weakly functorial balanced big Cohen-Macaulay algebra B_R and a blow-up Y \to Spec R for which the composition of the structure map with the natural map R \to RΓ(Y, O_Y) fails to vanish on the fiber in the derived category.","supporting_citations":[],"review_version":1}