{"id":"7b400199-6436-4817-ba36-47835c5b8f38","arxiv_id":"2509.06527","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Perfectoid towers over complete local domains yield lim Cohen-Macaulay sequences, and new perfectoid towers are constructed, including the first with p-torsion.","lead":"The paper shows that a complete local domain with a perfectoid tower automatically has a lim Cohen-Macaulay sequence, and it gives new recipes for building perfectoid towers from Frobenius-stable ideals. This connects two active tools in commutative algebra and offers a possible route toward Serre's positivity conjecture.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.19 invokes Definition 3.16 and Proposition 3.17 without proving that the perfectoid-tower layers R_i are finitely generated over R; Definition 2.3 does not supply this, so the main theorem needs an added finiteness/completeness hypothesis.","rationale":"The reader's weakest assumption is exactly the load-bearing problem I find. Re-reading the proof of Theorem 3.19, the only path to the conclusion is the reduction modulo f_0 via Proposition 3.17 and the comparison with the tilt; both routes require finite generation of the layers over R. Definition 2.3 does not provide it, and the proof neither cites Lemma 2.7 nor otherwise verifies the hypothesis. The same issue affects Corollary 3.18. I do not see a demonstrated contradiction: the constructions in Section 4 appear to yield finite layers, so the gap is best read as a missing hypothesis or missing proof, not as a known counterexample. Therefore the reader's CONDITIONAL verdict remains appropriate, and no verdict adjustment is needed.","tokens_in":26793,"tokens_out":20068,"duration_ms":249228,"concrete_test":"Check whether Definition 2.3, or the original definition in [18], implies that each layer R_i is I_0-adically complete and separated. Concretely, over R=Z_p[[x]] with I_0=(p), compare the standard completed tower S_i=V_i[[x^{1/p^i}]] (V_i=Z_p[p^{1/p^i}]) with the non-completed localized polynomial variant S'_i=V_i[x^{1/p^i}]_{m_i}. Verify which axioms of Definition 2.3 the variant satisfies. If it satisfies them, then S'_1 is not finite over R and Theorem 3.19 is false as stated; if it fails, identify the failing axiom and insert that condition into Theorem 3.19.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing reduction in Theorem 3.19 passes from {R_i} to {R_i/f_0R_i} via Proposition 3.17 and then compares with the tilt. But Definition 3.16 defines a lim Cohen-Macaulay sequence only for finitely generated R-modules, and Proposition 3.17 has the same hypothesis. Nowhere in Definition 2.3 is R_i required to be module-finite over R_0=R, and the proof of Theorem 3.19 does not prove it. Lemma 2.7 gives module-finiteness of the transition maps only under the extra hypothesis that each R_i is I_0-adically complete and separated; axiom (e) merely says I_0-adically Zariskian. (The F-finite hypothesis on R^flat_0 is automatic for a perfect tower, so completeness is the missing ingredient.) Thus, on the face of the axioms, a perfectoid tower with non-finite layers would make Theorem 3.19 undefined rather than proven, and even for finite layers the proof omits the required verification. The same gap appears in Corollary 3.18: finiteness of R_i/pR_i and equality of Krull dimensions do not by themselves imply R_i is finite over R. This is a genuine correctness risk for the central claim, though it is fixable by adding an explicit finite-generation or completeness hypothesis, or by proving that it follows from the cited definition in [18].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops connections among δ-rings, perfectoid towers, and lim Cohen-Macaulay sequences. Its central result, Theorem 3.19, asserts that any perfectoid tower over a complete Noetherian local domain of mixed characteristic with perfect residue field yields a lim Cohen-Macaulay sequence of algebras. The proof proceeds by reducing modulo a generator of the defining ideal, comparing with the tilt, and invoking results of Bhatt–Hochster–Ma on perfect towers. Sections 3 and 4 also study δ-rings and Frobenius lifts, introduce φ-stable ideals, and give constructions of perfectoid towers from monomial, binomial, and determinantal ideals, from section rings of smooth projective varieties, and from a new example with p-torsion elements.","tokens_in":27197,"tokens_out":13437,"duration_ms":161559,"significance":"If Theorem 3.19 is correct, it provides a broad new source of lim Cohen-Macaulay sequences and gives a concrete connection between perfectoid towers and Serre's positivity conjecture. The paper also contains useful explicit constructions, including the first example of a perfectoid tower with p-torsion elements, and a geometric construction via section rings. The main weakness is a missing finite-generation verification in the proof of the main theorem; this is a load-bearing gap, but it appears fixable by adding the missing hypotheses or proving finiteness from the cited definitions.","major_comments":[{"comment":"Definition 3.16 defines a lim Cohen-Macaulay sequence only for finitely generated R-modules. The proof of Theorem 3.19 applies Proposition 3.17 to the sequence {R_i} without proving that each R_i is finitely generated over R. Definition 2.3 does not include this property: axiom (e) only says R_i is I0-adically Zariskian. Lemma 2.7 gives module-finiteness of the transition maps only under extra hypotheses (F-finite tilt and I0-adic completeness/separatedness of each R_i), neither of which is assumed in Theorem 3.19. Thus on the stated hypotheses the object named in the conclusion may not even satisfy Definition 3.16. The same gap affects Corollary 3.18, where finiteness of R_i/pR_i and equality of Krull dimensions do not imply R_i is finite over R without additional completeness or finite-generation information. The theorem should be repaired by adding an explicit finite-generation/comple","section":"Theorem 3.19, Definition 3.16, Lemma 2.7"},{"comment":"After reducing modulo f0, the proof asserts that {R_i/f0R_i} is lim Cohen-Macaulay if and only if {R_i^♭/f0^♭R_i^♭} is, based on the ring isomorphisms R_i^♭/f0^♭R_i^♭ ≅ R_i/f0R_i. However Proposition 3.17 is stated for sequences of finitely generated modules over a fixed base local ring. The proof does not identify the base rings R/(f0) and R^♭_0/(f0^♭), nor does it check that either sequence satisfies the finite-generation hypothesis required by Proposition 3.17. The first point is likely repairable because R^♭_0/(f0^♭) ≅ R/(f0) by construction of the small tilt, but the second is the same finiteness gap as in the previous comment.","section":"Theorem 3.19, tilt comparison"},{"comment":"The definition of A_i (and R_i) as a 'finite colimit' of a diagram A --φ--> A --φ--> ... --φ--> A is literally the last copy of A, so with the written definition A_i = A for all i. This is inconsistent with later claims such as A_i ≅ W(k)[[x_1^{1/p^i}, ..., x_d^{1/p^i}]] in Discussion 4.7 and the computation of R_i in Corollary 4.8. The authors presumably intend the i-th stage of the direct limit, i.e. the subring of A^{1/p^∞} generated by the φ^i-th roots of elements of A, or an equivalent formulation. This is not merely a typo, because Corollary 3.18 and the constructions in Section 4 rely on the explicit form of R_i as a p^{1/p^i}-root extension.","section":"Definition 3.2, Definition 4.1, Discussion 4.7"}],"minor_comments":[{"comment":"Several cross-references use 'Theorem' where 'Lemma' or 'Definition' is meant, e.g. 'Theorem 3.4' in the proof of Lemma 3.4, 'Theorem 2.5 (3)' for Lemma 2.5, 'Theorem 3.13 (2)' for Lemma 3.13, and 'Theorem 4.17' for Lemma 4.17. Please correct.","section":"Throughout"},{"comment":"Typo in heading: 'determiantal' should be 'determinantal'.","section":"Section 4.2 heading"},{"comment":"The proof says the tower satisfies 'conditions (i), (ii), (iii) in Theorem 4.6', but Theorem 4.6 states only conditions (i) and (ii). Clarify the reference.","section":"Theorem 4.18(1)"},{"comment":"In the definition, ℓ_R(H_i(x;M_n)) uses the length over R, but the condition is written with a system of parameters of R. It would be helpful to state explicitly that the same system of parameters is used for all n and that the length is taken after viewing M_n as an R-module via the structure map.","section":"Definition 3.16"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a promising core idea, and the examples are valuable. The central theorem, however, is currently not proven as stated because the finite-generation condition required by the definition of lim Cohen-Macaulay sequences is not established. The issue is likely repairable by adding explicit completeness/F-finiteness hypotheses, but the statement then needs to be modified accordingly. The paper is close in spirit to work of Ishizuka [19] and the authors acknowledge this; the present contribution adds the lim Cohen-Macaulay connection and new examples with p-torsion. I recommend major revision rather than rejection because the main construction is plausible and the gaps are identifiable and local."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's main claim, Theorem 3.19, is a real connection: if a complete mixed-characteristic local domain admits a perfectoid tower, then the tower layers form a lim Cohen-Macaulay sequence. That would give a new route toward Serre's positivity conjecture. The idea of passing to the tilt and using the known lim CM property of perfect towers is natural and attractive. The paper is also honest about what it cannot do, for instance Section 4.4 notes that its p-torsion examples do not fall under Theorem 3.19.\n\nThe soft spot is real and load-bearing. Definition 3.16 requires the modules to be finitely generated over R, but the axioms for a perfectoid tower in Definition 2.3 do not include finite generation of the layers R_i over R_0. The proof of Theorem 3.19 does not supply it, and Lemma 2.7 only gives module-finiteness under extra hypotheses (I_0-adic completeness and F-finite tilt). Without that, the conclusion of Theorem 3.19 may be undefined rather than proven. The same issue appears in Corollary 3.18, where finiteness of R_i/pR_i does not imply finiteness of R_i over R. This is fixable, either by adding a finite-generation or completeness hypothesis to the standing assumptions, or by proving it follows from the cited definition in [18]. But as written, the proof is incomplete.\n\nThe positive side: the paper does useful work. The construction in Theorem 4.6, using phi-stable ideals and Frobenius lifts, gives concrete perfectoid towers for monomial and binomial quotients; the authors acknowledge overlap with Ishizuka [19] and the proof difference is real. The p-torsion examples in Section 4.4 look new. The citations to [18] are to prior work that supplies the infrastructure, and I see no circularity. The geometric section depends on an unpublished companion paper [21], so those examples are conditional.\n\nWho benefits? Researchers working on mixed-characteristic commutative algebra, lim Cohen-Macaulay sequences, perfectoid towers, or Serre's positivity. The paper deserves a careful referee. The central idea is good, and the gap is the kind that can be closed with an added hypothesis or a short proof. I'd send it to review, with a request to verify finite generation in Theorem 3.19 and Corollary 3.18 before publication.","headline":"The core theorem connecting perfectoid towers to lim Cohen-Macaulay sequences is plausible and worth serious attention, but the proof has a gap: it never shows the tower layers are finitely generated over R, as Definition 3.16 requires.","tokens_in":27626,"tokens_out":1720,"would_cite":false,"duration_ms":23362,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A35","13D22","13F55","13F65","14G45"],"pacs":[],"model":"deepseek-v4-flash","headline":"A complete mixed-characteristic local domain admitting a perfectoid tower has a lim Cohen–Macaulay sequence, linking tilting to the positivity conjecture.","keywords":["δ-ring","Frobenius lift","perfectoid tower","lim Cohen-Macaulay sequence","φ-stable ideal","mixed characteristic commutative algebra","perfectoid purity","intersection multiplicity positivity conjecture"],"falsifier":"A perfectoid tower satisfying Definition 2.3 whose tilt is not F-finite, or with a transition map R_0 → R_1 that is not module-finite, would leave Theorem 3.19 unusable: Definition 3.16 requires each layer to be a finitely generated R-module. Concretely, one could look for a tower over R = W(k)[[x_2,...,x_d]]/I with squarefree monomial I in which the colimit layer R_1 is not finite as an R-module; if such a tower exists, the conclusion 'lim Cohen–Macaulay sequence' has no meaning under the paper's own definition.","tokens_in":26756,"feed_emoji":"🧮","tokens_out":11397,"duration_ms":105070,"temperature":0.7,"pith_summary":"This paper attempts to connect two objects in commutative algebra: perfectoid towers and lim Cohen–Macaulay sequences. A perfectoid tower is a Noetherian approximation of a perfectoid ring arranged as a chain of ring maps whose reductions are purely inseparable; a lim Cohen–Macaulay sequence is a sequence of finitely generated modules whose local cohomology lengths grow slowly with the number of generators, a substitute for a Cohen–Macaulay module. The main theorem says that whenever a complete Noetherian local domain of mixed characteristic with perfect residue field admits a perfectoid tower, its layers form a lim Cohen–Macaulay sequence. The proof passes to the tilt, where the tower becomes a perfect tower of p-power roots, and perfect towers were already known to be lim Cohen–Macaulay. The paper then supplies construction methods for perfectoid towers from δ-rings and stable Frobenius lifts, with explicit examples from monomial, binomial, determinantal, and section rings; if the main theorem is right, the existence of such towers would give a route to the positivity conjecture on intersection multiplicities.","feed_headline":"Perfectoid towers yield lim Cohen-Macaulay sequences","feed_subtitle":"Every perfectoid tower over a mixed-characteristic local domain gives the module sequences needed for the positivity conjecture.","key_machinery":"The carrying mechanism is the tilting correspondence for perfectoid towers. From a tower satisfying the axioms, one takes the inverse limit of the layers modulo I_0 along the Frobenius projections; the result is a perfect tower, naturally isomorphic to adjoining successive p-power roots of the small tilt. The isomorphism R_i/f_0R_i ≅ R_i^♭/f_0^♭ R_i^♭ lets the lim Cohen–Macaulay property travel from the perfect tower back to the original one. On the construction side, the central object is a φ-stable ideal in a δ-ring: a ring A with a p-derivation δ whose associated Frobenius lift φ satisfies φ(I) ⊆ I. An ideal with this closure property lets the quotient R=A/I inherit a Frobenius lift, and","core_discovery":"The paper's central claim is Theorem 3.19: let (R,m,k) be a complete Noetherian local domain of mixed characteristic with perfect residue field of characteristic p. If a perfectoid tower ({R_i},{t_i}) arises from (R,I_0) for some ideal I_0, then the sequence {R_i} is a lim Cohen–Macaulay sequence of algebras. The proof fixes a generator f_0 of I_0 and uses the quoted facts that the I_0-torsion of each R_i vanishes and that the quotient by f_0 transfers the lim Cohen–Macaulay property. Because the reduced quotients R_i/f_0R_i are isomorphic to the corresponding quotients of the tilt R_i^♭/f_0^♭ R_i^♭, and because the tilt is a perfect tower isomorphic to R_0^♭ → (R_0^♭)^{1/p} → ..., already l","pith_inferences":["Editorial inference: The finite-generation gap in Theorem 3.19 suggests that the natural strengthening replaces 'perfectoid tower exists' by 'perfectoid tower with F-finite tilt exists'; one could test whether the geometric examples of Section 4.3 satisfy this stronger hypothesis.","Editorial inference: The φ-stable ideal method is not tied to formal power series; any δ-ring quotient with a φ-stable ideal and reduced p-reduction should yield a perfectoid tower. A testable extension would be to determinantal ideals of k×m-minors for k>2 or to non-ladder binomial ideals, where φ-stability becomes a combinatorial condition on exponents.","Editorial inference: The p-torsion example in Section 4.4 is explicitly outside Theorem 3.19; a generalized definition of lim Cohen–Macaulay sequence allowing torsion or non-finite layers might recover a theorem for such towers and enlarge the supply of examples.","Editorial inference: Since the final step of the proof imports [3]'s theorem on perfect towers, any future strengthening of that theorem would automatically strengthen Theorem 3.19; conversely, a counterexample to that theorem would not necessarily invalidate the construction parts of the paper."],"forward_implications":["If Theorem 3.19 holds, constructing a perfectoid tower over any complete Noetherian local domain of mixed characteristic would produce the lim Cohen–Macaulay sequence needed for the positivity conjecture on intersection multiplicities, making perfectoid-tower existence the central question.","The φ-stable ideal construction in Theorem 4.6 shows that quotients of W(k)[[x]] by monomial or binomial ideals, and by ideals of 2×m-minors, carry perfectoid towers whenever they are p-torsion-free with reduced reduction, giving explicit lim Cohen–Macaulay sequences.","For section rings of smooth projective varieties with quasi-canonical liftings, the construction yields perfectoid towers whose layers are complete normal local domains; when the variety is an ordinary abelian variety of dimension at least 2, each layer is non-Cohen–Macaulay, so these are genuinely new lim Cohen–Macaulay examples.","In the p-torsion-free cases the tilt is explicitly the Frobenius-root tower over the residue field (Theorem 4.6(2), Corollaries 4.8 and 4.10), so the Frobenius structure of the tower is completely computable.","Lemma 2.10 and Corollary 4.11 turn perfectoid towers with pure transition maps into perfectoid purity of the base ring, yielding new perfectoid pure singularities such as lifts of wide Gorenstein ladder determinantal rings."],"supporting_citations":[{"why":"Supplies the definition of perfectoid towers, their tilts, and the transfer properties (torsion, Noetherianity, completion) used throughout.","marker":"[18]"},{"why":"Defines lim Cohen–Macaulay sequences and proves that perfect Frobenius-root towers are lim Cohen–Macaulay, the terminal step of Theorem 3.19.","marker":"[3]"},{"why":"Formulates the conjecture that every complete local domain has a lim Cohen–Macaulay sequence and explains that it implies the positivity conjecture on intersection multiplicities.","marker":"[16]"},{"why":"Provides the δ-ring and Frobenius-lift formalism plus prisms, the algebraic raw material for the tower constructions in Section 4.","marker":"[5]"},{"why":"Gives a prior construction of perfectoid towers from prisms that the paper's φ-stable ideal construction parallels and extends.","marker":"[19]"},{"why":"Supplies the quasi-canonical liftings of projective varieties used to build the geometric perfectoid towers in Section 4.3.","marker":"[21]"},{"why":"Provides the notion of perfectoid purity used in Corollary 4.8(3) and for the perfectoid injectivity criterion in Section 4.3.","marker":"[7]"},{"why":"Shows the completed section rings of abelian varieties have maximal Cohen–Macaulay modules and fail to be Cohen–Macaulay, supporting the non-CM examples.","marker":"[34]"}],"fun_headline_variants":["Perfectoid towers forge lim Cohen-Macaulay sequences","Perfectoid towers deform purity into lim CM","From perfectoid towers to lim Cohen-Macaulay","Perfectoid towers enable lim Cohen-Macaulay construction","Perfectoid towers: building blocks for lim CM"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The conclusion only makes sense if every layer R_i is finitely generated as an R-module, but the axioms of a perfectoid tower do not guarantee finite generation; the paper obtains it only under extra hypotheses such as an F-finite tilt and I_0-adic completeness.","fun_headline_variants_meta":{"raw":{"variants":["Perfectoid towers forge lim Cohen-Macaulay sequences","Perfectoid towers deform purity into lim CM","From perfectoid towers to lim Cohen-Macaulay","Perfectoid towers enable lim Cohen-Macaulay construction","Perfectoid towers: building blocks for lim CM"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000735,"raw_usage":{"total_tokens":3098,"prompt_tokens":695,"completion_tokens":2403,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":439,"completion_tokens_details":{"reasoning_tokens":2327}},"tokens_in":439,"tokens_out":2403,"duration_ms":20990,"temperature":1.0,"reasoning_tokens":2327,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T23:28:47.786559+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A perfectoid tower satisfying Definition 2.3 whose tilt is not F-finite, or with a transition map R_0 → R_1 that is not module-finite, would leave Theorem 3.19 unusable: Definition 3.16 requires each layer to be a finitely generated R-module. Concretely, one could look for a tower over R = W(k)[[x_2,...,x_d]]/I with squarefree monomial I in which the colimit layer R_1 is not finite as an R-module; if such a tower exists, the conclusion 'lim Cohen–Macaulay sequence' has no meaning under the paper's own definition.","supporting_citations":[{"cited_title":"Ishiro, K","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of perfectoid towers, their tilts, and the transfer properties (torsion, Noetherianity, completion) used throughout."},{"cited_title":"Bhatt, M","cited_arxiv_id":null,"evidence_quote":"Defines lim Cohen–Macaulay sequences and proves that perfect Frobenius-root towers are lim Cohen–Macaulay, the terminal step of Theorem 3.19."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Formulates the conjecture that every complete local domain has a lim Cohen–Macaulay sequence and explains that it implies the positivity conjecture on intersection multiplicities."},{"cited_title":"Bhatt and P","cited_arxiv_id":null,"evidence_quote":"Provides the δ-ring and Frobenius-lift formalism plus prisms, the algebraic raw material for the tower constructions in Section 4."},{"cited_title":"Ishizuka,Perfectoid towers generated from prisms,https://arxiv.org/abs/2409.15785v2","cited_arxiv_id":null,"evidence_quote":"Gives a prior construction of perfectoid towers from prisms that the paper's φ-stable ideal construction parallels and extends."},{"cited_title":"Ishizuka and K","cited_arxiv_id":null,"evidence_quote":"Supplies the quasi-canonical liftings of projective varieties used to build the geometric perfectoid towers in Section 4.3."},{"cited_title":"Shimomoto and E","cited_arxiv_id":null,"evidence_quote":"Shows the completed section rings of abelian varieties have maximal Cohen–Macaulay modules and fail to be Cohen–Macaulay, supporting the non-CM examples."}],"review_version":1}