{"id":"a4a3ae85-cbb4-4e09-b59e-0a38f5291f8e","arxiv_id":"2506.00529","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Associate primes, grades, and finite lengths of F(M/I_1^{n_1}...I_r^{n_r}N) are eventually stable or polynomial, with a sharp degree bound.","lead":"The paper proves that several algebraic measurements of modules built from powers of multiple ideals become stable or polynomial once the exponents are large. It unifies and extends earlier one-variable results to a general class of functors, with applications to Betti and Bass numbers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.6's reduction hinges on a strong product-form multigraded Artin-Rees identity and on [10, Prop. 3.4]; neither is verified in the text, and if either fails the central stability theorem does not follow.","rationale":"The reader's weakest assumption points to the strong multigraded Artin-Rees identity, and I agree that this is the most load-bearing step in the reduction. However, a fuller reading shows that the identity is very likely derivable from the standard multigraded Artin-Rees recurrence plus finite generation of the Rees submodule, so the concern is not a demonstrated counterexample but an unproven step in the text. A second, comparably load-bearing import is [10, Prop. 3.4], used without stating its hypotheses; if its hypotheses are narrower than the application, Theorem 2.12(1)-(2) would not be established. The rest of the paper—Theorem 2.3, the polynomial behavior arguments, and the applications to Betti/Bass numbers and homological dimensions—appears internally consistent given these two imports. The reader's CONDITIONAL verdict is therefore appropriate: no change to the verdict is needed, but the requested revisions should include an explicit proof or precise citation for the product-form Artin-Rees identity and a statement of [10, Prop. 3.4] with verification that it covers the multivariable, arbitrary-Noetherian-ring setting.","tokens_in":12497,"tokens_out":40615,"duration_ms":356435,"concrete_test":"Read [16, 17.1.6] and [10, Prop. 3.4] in the original sources. If [16, 17.1.6] states only the componentwise recurrence, write out the finite-generation argument showing ψ(B) ∩ I^nC' = I^{n-d}(ψ(B) ∩ I^dC') for d chosen larger than the generator degrees of ⊕_n(ψ(B) ∩ I^nC') in ⊕_n I^nC', and check whether this argument uses any hidden hypotheses (e.g., ideals generated by regular sequences or local rings). Independently, verify that [10, Prop. 3.4] applies verbatim to (U + I_1^{n_1}...I_r^{n_r}V)/I_1^{n_1}...I_r^{n_r}W over arbitrary Noetherian R with W ⊆ V; if the proposition is only for r=1 or for local R, the reduction in Theorem 2.12(1)-(2) needs an additional proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 2.5, the identity ψ(B) ∩ I^nC' = I^{n-d}(ψ(B) ∩ I^dC') is used to derive ψ^{-1}(I^nC') = ker ψ + I^{n-d}ψ^{-1}(I^dC'), and this is what converts F(M/I^nN) into the module (U + I^{n-d}V)/I^{n-d}W. The cited [16, 17.1.6] is the multigraded Artin-Rees lemma; its standard statement is a componentwise recurrence, N ∩ I^nM = Σ_i I_i(N ∩ I^{n-e_i}M) for n ≥ d, not the single-product form used here. The product form can be deduced if the Rees submodule ⊕_n (ψ(B) ∩ I^nC') is generated in degrees ≤ d, but the paper does not supply this argument. Similarly, Theorem 2.12(1) and (2) import [10, Prop. 3.4] verbatim to conclude that Ass_R((U+I^nV)/I^nW) stabilizes; if that proposition is only proved for a single ideal, for local rings, or under other extra hypotheses, the proof of the main theorem is incomplete. These two imports are the load-bearing pillars: the first connects coherent functors to the family (U+I^nV)/I^nW, and the second gives the asymptotic stability of that family.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for a coherent functor F on the category of finitely generated modules over a Noetherian ring R, the sets Ass_R(F(M/I_1^{n_1}...I_r^{n_r}N)) and grade(J, F(M/I_1^{n_1}...I_r^{n_r}N)) stabilize for large multi-indices, and that if F(M/I_1^{n_1}...I_r^{n_r}N) has finite length then its length is eventually given by a multivariable polynomial. In the local case, a degree bound max{dim F(M), ℓ_M(I)-r} is proved, with equality when dim F(M) > ℓ_M(I)-r. Applications include eventual polynomiality of Betti and Bass numbers and constancy of projective and injective dimension.","tokens_in":12802,"tokens_out":47282,"duration_ms":441329,"significance":"If correct, the main theorem is a substantial unification of earlier results by Brodmann, Kingsbury-Sharp, Katz-West, Kodiyalam, Theodorescu, Se, and Banda-Melkersson, covering arbitrary products of ideals and all coherent functors. The proof strategy, reducing F(M/I^nN) to a module of the form (U+I^{n-d}V)/I^{n-d}W, is elegant, and the paper includes self-contained proofs of some auxiliary reductions (Lemmas 2.4, 2.5 and Proposition 2.7). The applications to Betti and Bass numbers, with degrees independent of i and of F, are valuable. However, the manuscript currently contains a defective short exact sequence in Theorem 2.9 and an insufficiently justified strong form of the multigraded Artin-Rees lemma; both are load-bearing steps in the proof. The issues appear local and fixable, so the central claims are likely correct.","major_comments":[{"comment":"The displayed sequence 0 → ((U+I^nW) ∩ I^nV)/I^nW → (U+I^nW)/I^nW → L_n → 0 is not a short exact sequence. The map from the middle term to L_n is the natural inclusion, whose kernel is zero and whose cokernel is (U+I^nV)/(U+I^nW), not zero. Hence the stated inference that λ_R(L_n) is a polynomial once the first two modules are polynomial does not follow from this sequence. Please replace (2.6) with a correct exact sequence, for example 0 → I^nV/I^nW → L_n → U/(U∩I^nV) → 0, and adjust the degree-bound argument accordingly. This is a load-bearing error because Theorem 2.9 is used in the proof of Theorem 2.12(3).","section":"Theorem 2.9, Eq. (2.6)"},{"comment":"The identity ψ(B) ∩ I^nC' = I^{n-d}(ψ(B) ∩ I^dC') is asserted as a direct consequence of [16, 17.1.6]. The standard multigraded Artin-Rees lemma gives a componentwise sum (N ∩ I^nM = Σ_i I^{n-e_i}(N ∩ I^{e_i}M) in the usual formulation), not the displayed single-product form. The single-product form is true—it follows by choosing d larger than the degrees of a generating set of the Rees module ⊕_n (ψ(B) ∩ I^nC')—but this justification is not supplied in the manuscript. Since this identity is the mechanism that converts F(M/I^nN) into (U+I^{n-d}V)/I^{n-d}W, the proof should include the missing argument or cite a source that states precisely this strong form.","section":"Lemma 2.5 and Theorem 2.6"},{"comment":"The proof imports [10, Prop. 3.4] to conclude that Ass_R((U+I^nV)/I^nW) stabilizes. The proposition is not stated in the manuscript, and its hypotheses are not verified against the present multigraded, non-local, arbitrary Noetherian ring setup. This is a load-bearing pillar for both the Ass and grade stabilization statements. Please quote the proposition in full, including its hypotheses, or provide a self-contained proof.","section":"Theorem 2.12(1)–(2)"}],"minor_comments":[{"comment":"The arXiv header shows 'ST ABILITY' instead of 'STABILITY'; please fix the spacing.","section":"Title"},{"comment":"When claiming that ⊕_n F(M_n) is a quotient of ⊕_n h_K(M_n), the finite generation of ⊕_n h_K(M_n) over S is used implicitly; this follows by taking a presentation of K, but it should be stated for completeness.","section":"Theorem 2.3, proof"},{"comment":"The phrase 'highly generalizes' is informal; please specify in what precise sense Theorem 2.10 extends [17, Cor. 4].","section":"Remark 2.11"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the novelty is adequate. The main concern is verifiability: two external results ([16, 17.1.6] in a strong form and [10, Prop. 3.4]) are used without statement or proof, and one internal short exact sequence, Eq. (2.6), is simply wrong. I believe the central claims are correct and the issues are fixable, so I recommend major revision rather than rejection. The authors should especially repair Eq. (2.6), as it is a clear mathematical error in the proof of Theorem 2.9."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper completes a natural program. Se and Banda-Melkersson handled coherent functors for powers of a single ideal; Kingsbury-Sharp, Katz-West, and West handled products of powers for modules and for Ext/Tor. Here the authors prove the full multivariable statement for any coherent functor: Ass and grade stabilize for F(M/I^nN), the length is eventually a polynomial, and in the local case the degree is sharply bounded by max{dim F(M), l_M(I)-r}. That is the theorem people have been expecting, and the proof is put together cleanly.\n\nThe architecture is sound. Lemma 2.5 converts F(M/I^nN) into (U + I^{n-d}V)/I^{n-d}W using the product-form multigraded Artin-Rees lemma, and Theorem 2.9 extracts the polynomial and degree bound via Hilbert functions and a determinant trick. Theorem 2.10's sharp bound is the genuinely new technical content; the equality case is a nice touch. The applications to Betti and Bass numbers and constancy of projective/injective dimension are straightforward but useful.\n\nThe soft spots are mainly cosmetic, plus one thing to check. The diagrams in Theorem 2.6 are garbled in the arXiv rendering, and Theorem 3.1's notation clashes with the earlier Rees ring notation. More substantively, the proof leans on two imported results: the multigraded Artin-Rees identity in Lemma 2.5 and [10, Prop. 3.4] for stabilization of Ass((U+I^nV)/I^nW). The first is indeed the product form in Swanson-Huneke 17.1.6, so the stress-test concern there is overstated. A referee should still confirm that [10, Prop. 3.4] is stated in the full generality needed (arbitrary Noetherian ring, several ideals, arbitrary submodules). If it is only for local rings or a single ideal, the proof of Theorem 2.12 needs a patch. I suspect it is fine, but it should be spelled out.\n\nOverall: solid, useful, and complete enough to deserve serious refereeing. I would send it to a commutative algebra journal with a request to clean up the diagrams and to state the imported propositions explicitly.","headline":"Clean completion of the coherent-functor asymptotic program for products of powers of several ideals, with sharp degree bounds; the main proofs are sound and the only real issues are presentation and two imported results worth verifying.","tokens_in":13375,"tokens_out":4170,"would_cite":true,"duration_ms":39854,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D07","13A15","13A02","13D02"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that coherent functors make the associated primes, grades, and lengths of modules of the form $M/(I_1^{n_1}\\cdots I_r^{n_r}N)$ stabilize or become polynomial as the exponents grow, with a sharp degree bound in the local…","keywords":["coherent functors","asymptotic stability","associate primes","grade","multigraded Hilbert polynomials","Betti numbers","Bass numbers","powers of ideals"],"falsifier":"Compute the invariants for a concrete coherent functor such as $\\operatorname{Tor}_1^R(k,-)$ on a local ring $R$ with two ideals $I_1, I_2$, taking $M=R$ and $N=0$: the theorem predicts that $\\operatorname{Ass}_R(\\operatorname{Tor}_1^R(k, R/(I_1^{n_1}I_2^{n_2})))$ stabilizes and, whenever lengths are finite, they match a polynomial in $(n_1,n_2)$ for large exponents. An explicit oscillation or a non-polynomial finite-length sequence would disprove Theorem 2.12; equivalently, checking the multigraded Artin–Rees identity on that example would locate the failure.","tokens_in":12319,"feed_emoji":"🧮","tokens_out":7443,"duration_ms":85689,"temperature":0.7,"pith_summary":"The paper establishes a common generalization of several asymptotic stability results: when $F$ is a coherent functor on finitely generated modules over a Noetherian ring, applying $F$ to quotients $M/(I_1^{n_1}\\cdots I_r^{n_r}N)$ yields associated prime sets and grades that eventually stop changing. It also proves that if these modules have finite length for all large exponent vectors, then the length is eventually given by a polynomial in the $r$ exponent variables. In the local case, it gives a sharp upper bound on the total degree of that polynomial, namely $\\max\\{\\dim(F(M)), \\ell_M(I)-r\\}$, with equality when $\\dim(F(M)) > \\ell_M(I)-r$. Because $\\operatorname{Tor}$ and $\\operatorname{Ext}$ functors are coherent, the same theorem yields eventual polynomiality of Betti and Bass numbers and constancy of projective and injective dimension. The result unifies and extends earlier single-ideal and single-functor results into one framework.","feed_headline":"Coherent functors turn ideal-power invariants into polynomials","feed_subtitle":"One theorem covers prime sets, grades, lengths, Betti and Bass numbers across products of r ideal powers.","key_machinery":"A coherent functor is a covariant $R$-linear functor that fits in an exact sequence $h_K \\to h_M \\to F \\to 0$ of Hom functors; this class includes $\\operatorname{Tor}$, $\\operatorname{Ext}$, and homology functors of tensor complexes. The load-bearing construction is Theorem 2.6: after lifting a presentation of $F$ to free resolutions and applying the multigraded Artin–Rees lemma, every value $F(M/I^{\\underline{n}}N)$ is exhibited as $(U + I^{\\underline{n}-d}V)/I^{\\underline{n}-d}W$, with $U \\cong F(M)$ and $W \\subseteq V$. This single normal form converts questions about all exponent vectors into questions about a fixed finite module $U$ and a product-ideal filtration, which is exactly what the multigraded Hilbert polynomial and degree estimates can answer.","core_discovery":"The central discovery is that every coherent functor $F$ admits a stable finite presentation on the family $M/I^{\\underline{n}}N$: for some exponent vector $d$, one has $F(M/I^{\\underline{n}}N) \\cong (U + I^{\\underline{n}-d}V)/I^{\\underline{n}-d}W$ for all $\\underline{n} \\geq d$, where $U \\cong F(M)$ and $W \\subseteq V$. Once this normal form is in hand, the eventual stabilization of $\\operatorname{Ass}_R$ and $\\operatorname{grade}$, the polynomial growth of length, and the degree bounds all follow from standard facts about multigraded modules and their Hilbert polynomials. Thus the paper reduces the entire problem to showing that coherent functors carry asymptotic stability from the ambient ring to arbitrary derived operations, uniformly in $r$ ideals.","pith_inferences":["Because the class of coherent functors is closed under composition, the same arguments should give stability and polynomiality for iterated coherent functors, not just a single $F$; this would extend the theorem to repeated derived operations without new ideas.","The degree bound is independent of $N$ and of the index $i$, which suggests that the leading asymptotic term of the length polynomial may be governed entirely by the pair $(F(M), \\ell_M(I))$; testing this on concrete examples would be a natural numerical experiment.","The equality condition $\\dim(F(M)) > \\ell_M(I)-r$ separates growth coming from the functor's target dimension from growth coming from the ambient ideal filtration, so algorithms that compute $\\ell_M(I)$ could turn the theorem into explicit stabilization thresholds."],"forward_implications":["If $F$ is coherent and $F(M/I^{\\underline{n}}N)$ has finite length for all large $\\underline{n}$, then the length function is eventually a polynomial over $\\mathbb{Q}$ in the $r$ variables $n_1,\\ldots,n_r$.","In the local case, the total degree of that polynomial is at most $\\max\\{\\dim(F(M)), \\ell_M(I)-r\\}$, and equality holds whenever $\\dim(F(M)) > \\ell_M(I)-r$, giving a computable growth rate.","For each fixed $i$, the $i$th Betti number $\\beta_i^R(F(M/I^{\\underline{n}}N))$ and the $i$th Bass number $\\mu^i_R(F(M/I^{\\underline{n}}N))$ are eventually polynomial in $\\underline{n}$, with degree bounded by $\\max\\{0, \\ell_M(I)-r\\}$.","Consequently, projective dimension and injective dimension of $F(M/I^{\\underline{n}}N)$ are eventually constant for large $\\underline{n}$.","The same machinery also applies to the graded components $\\mathcal{M}_{\\underline{n}}$ of a finitely generated multigraded module over a Noetherian standard $\\mathbb{N}^r$-graded ring, so both natural families of modules are covered."],"supporting_citations":[{"why":"Supplies the multigraded Artin–Rees lemma used to produce the normal form $(U + I^{\\underline{n}-d}V)/I^{\\underline{n}-d}W$.","marker":"[16]"},{"why":"Provides the stabilization result for $\\operatorname{Ass}_R((U + I^{\\underline{n}}V)/I^{\\underline{n}}W)$, the engine behind the stability statements in Theorem 2.12.","marker":"[10]"},{"why":"Supplies the multigraded Hilbert-polynomial theorem used to obtain polynomiality and degree bounds for the filtered modules.","marker":"[9]"},{"why":"Proves the single-ideal coherent-functor asymptotic stability results that this paper extends to several ideals.","marker":"[15]"},{"why":"Gives the coherent-functor toolkit, including closedness under composition, which is needed for the $\\operatorname{Ext}$ and $\\operatorname{Tor}$ applications.","marker":"[2]"},{"why":"Proves derived-functor Hilbert polynomials with sharp degree bounds; the proof of Lemma 2.5 follows its idea.","marker":"[17]"},{"why":"Shows stability of associated primes for quotients by products of powers, used at the start of the length-polynomial proof.","marker":"[11]"},{"why":"Supplies stabilization of associated primes and grades for multigraded modules, used for the graded-component statements.","marker":"[18]"}],"fun_headline_variants":["Polynomial invariants from coherent functors on ideal powers","Coherent functors force polynomial growth in powers","Stable invariants for large powers under coherent functors","Asymptotic polynomials for coherent functors on powers","Coherent functors: from powers to polynomials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the multigraded Artin–Rees identity—intersecting with a product of ideal powers eventually just shifts by a fixed power—which the paper imports rather than proves; if that identity fails for several ideals, the reduction of $F(M/I^{\\underline{n}}N)$ to a fixed quotient model collapses.","fun_headline_variants_meta":{"raw":{"variants":["Polynomial invariants from coherent functors on ideal powers","Coherent functors force polynomial growth in powers","Stable invariants for large powers under coherent functors","Asymptotic polynomials for coherent functors on powers","Coherent functors: from powers to polynomials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000332,"raw_usage":{"total_tokens":1989,"prompt_tokens":1229,"completion_tokens":760,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":845,"completion_tokens_details":{"reasoning_tokens":684}},"tokens_in":845,"tokens_out":760,"duration_ms":8636,"temperature":1.0,"reasoning_tokens":684,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:05:34.645478+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the invariants for a concrete coherent functor such as $\\operatorname{Tor}_1^R(k,-)$ on a local ring $R$ with two ideals $I_1, I_2$, taking $M=R$ and $N=0$: the theorem predicts that $\\operatorname{Ass}_R(\\operatorname{Tor}_1^R(k, R/(I_1^{n_1}I_2^{n_2})))$ stabilizes and, whenever lengths are finite, they match a polynomial in $(n_1,n_2)$ for large exponents. An explicit oscillation or a non-polynomial finite-length sequence would disprove Theorem 2.12; equivalently, checking the multigraded Artin–Rees identity on that example would locate the failure.","supporting_citations":[{"cited_title":"Integral closure of ideals, rings, and modules , volume 336 of Lond","cited_arxiv_id":null,"evidence_quote":"Supplies the multigraded Artin–Rees lemma used to produce the normal form $(U + I^{\\underline{n}-d}V)/I^{\\underline{n}-d}W$."},{"cited_title":"A linear function associated to asymptotic prime divisors","cited_arxiv_id":null,"evidence_quote":"Provides the stabilization result for $\\operatorname{Ass}_R((U + I^{\\underline{n}}V)/I^{\\underline{n}}W)$, the engine behind the stability statements in Theorem 2.12."},{"cited_title":"Reduction numbers and multiplicities of multi- graded structures","cited_arxiv_id":null,"evidence_quote":"Supplies the multigraded Hilbert-polynomial theorem used to obtain polynomiality and degree bounds for the filtered modules."},{"cited_title":"Covariant functors and asymptotic stability","cited_arxiv_id":null,"evidence_quote":"Proves the single-ideal coherent-functor asymptotic stability results that this paper extends to several ideals."},{"cited_title":"Coherent functors and asymptotic stability","cited_arxiv_id":null,"evidence_quote":"Gives the coherent-functor toolkit, including closedness under composition, which is needed for the $\\operatorname{Ext}$ and $\\operatorname{Tor}$ applications."},{"cited_title":"Derived functors and Hilbert polynomials","cited_arxiv_id":null,"evidence_quote":"Proves derived-functor Hilbert polynomials with sharp degree bounds; the proof of Lemma 2.5 follows its idea."},{"cited_title":"Kingsbury and Rodney Y","cited_arxiv_id":null,"evidence_quote":"Shows stability of associated primes for quotients by products of powers, used at the start of the length-polynomial proof."},{"cited_title":"Primes associated to multigraded modules","cited_arxiv_id":null,"evidence_quote":"Supplies stabilization of associated primes and grades for multigraded modules, used for the graded-component statements."}],"review_version":1}