{"id":"b056872d-6973-402c-b412-0f5b9ab768d6","arxiv_id":"2411.13090","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For polynomial rings, every nonzero local cohomology or related Lyubeznik module has nonzero pieces in all degrees at or below -m, and the top nonvanishing module has infinite-dimensional pieces.","lead":"Local cohomology modules are algebraic objects that measure how polynomial equations behave near their vanishing sets. This paper proves that, apart from two special cases, these modules have nonzero pieces in every sufficiently negative degree, and their top pieces are infinite-dimensional.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.1's final step is an unsupported and in general false inference: N free over S does not imply M free over R, and the proof of Theorem 1.2(1) relies on it.","rationale":"The reader's weakest_assumption pointed to the unproved structural theorems 2.6-2.8 quoted from the earlier papers, especially the unreviewed preprint [6]. That is a legitimate concern, and those theorems are indeed load-bearing for the main result. However, an even more immediate defect is the final inference in Lemma 3.1, which the reader's rationale mentioned but did not make the primary concern. The proof's transition from 'N is a free S-module' to 'M = R^s' is not valid in general, as the module (X1,X2) over K[X1,X2] shows. Since the counterexample is not a D-module, it does not refute the lemma as stated, but it does show that the burden is on the author to supply an argument using the D-module or F-finite structure. Without such an argument, the proof of Theorem 1.2(1) is incomplete, and Theorem 1.1 inherits that gap. I do not see a concrete counterexample to the central theorem itself, so rejection would be too strong; the appropriate disposition is conditional on a correct proof or replacement of Lemma 3.1. This largely agrees with the reader's conditional verdict but sharpens the reason: the weak point is the algebraic inference in Lemma 3.1, not only the dependence on unpublished structural results.","tokens_in":7464,"tokens_out":27025,"duration_ms":285164,"concrete_test":"Test the missing implication for the actual class of modules: prove or disprove that a graded generalized Eulerian holonomic A_m(K)-module M with M_n = 0 for n < 0 and with a regular homogeneous Xm such that M/XmM ≅ S^s must be free over R. A concrete way to run this test is to check whether M is generated by M0 as an R-module: if yes, the free quotient S^s forces M ≅ R^s; if no, exhibit a module with M_0 = 0 and a free quotient, which would make Lemma 3.1 false. This directly settles whether the unsupported inference in the proof of Lemma 3.1 is valid for the modules appearing in Theorems 1.1 and 1.2.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Lemma 3.1, after choosing a regular Xm and setting N = M/XmM, the induction hypothesis gives N ≅ S^s. The proof then asserts 'It follows that M = R^s'. For arbitrary graded modules this is false: with R = K[X1,X2] and M = (X1,X2), X1 is M-regular and N = M/X1M ≅ K[X2] is free over S = K[X2], but M is not free over R. This example is not a D-module, so it does not by itself disprove Lemma 3.1; however, it shows the written justification is not a valid consequence of the preceding facts. The paper supplies no argument using the F_R-finite or generalized Eulerian holonomic structure that would rule out such a module. Since Lemma 3.1 is used in Theorem 1.2(1) to infer that a nonzero J-torsion module T(R) must be R^s and hence impossible, the main nonvanishing conclusion rests on this missing step. This gap is independent of the status of Theorems 2.6-2.8 and must be repaired before the proof can be accepted.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies graded components of local cohomology modules H^i_I(R) for R = K[X_1,...,X_m] and homogeneous ideals I of height g with 1 ≤ g ≤ m−1. It claims that if H^i_I(R) is nonzero, then H^i_I(R)_n is nonzero for all n ≤ −m; if the support is not the maximal ideal, all graded components are nonzero (and in characteristic zero have infinite K-dimension); and that dim_K H^g_I(R)_n is infinite for all n. The proofs are framed through a general class of graded Lyubeznik functors T and rest on structural theorems quoted from the author's earlier work [5] and the unpublished preprint [6].","tokens_in":7722,"tokens_out":15583,"duration_ms":190714,"significance":"If correct, the results would be a substantial advance in understanding the sparsity of graded local cohomology modules, replacing scattered examples with a clean dichotomy: away from the maximal-ideal case, local cohomology modules over polynomial rings are either zero in a bounded range or nonzero in every sufficiently low degree, with infinite-dimensional components in characteristic zero. The paper is clearly written and the intended strategy—reduce to a general lemma about graded F-finite and generalized Eulerian holonomic modules—is natural. However, the central lemma on which all main theorems rest is false as stated, so the claimed results are not established by the arguments presented.","major_comments":[{"comment":"Lemma 3.1 is false as stated. Let R = K[X_1,...,X_m] with m ≥ 2 and char K = 0, and let M = (X_1,...,X_m). Then M is a nonzero graded generalized Eulerian holonomic A_m(K)-module (it is a D-submodule of R, hence holonomic, and E acts on degree d by multiplication by d). Since M has no components in degrees ≤ 0, the hypothesis M_n = 0 for n ≤ −1 holds. But M is not isomorphic to R^s for any s: it is not free, and in particular its degree-0 component is zero, while R^s has degree-0 component K^s. Thus the conclusion 'It follows that M = R^s' is impossible. The proof's final inference from finite generation, M/X_m M ≅ S^s, and X_m-regularity to freeness of M is invalid even without the D-module structure: with R = K[X_1,X_2], M = (X_1,X_2), and x = X_1, the module M is finitely generated and x-regular, N = M/xM ≅ K[X_2] is free over S, but M is not free. The proof uses no argument specific to F_R-finite or generalized Eulerian holonomic modules that would exclude such behaviour. Since Lemma 3.1 is applied in the proofs of Theorem 1.2(1), Theorem 1.2(2), and Theorem 1.1(3), the main conclusions of the paper are not supported.","section":"Lemma 3.1"},{"comment":"The statement in 2.9(2) that for m = 1 the modules H_l(X_1; M) are concentrated in degree 0 is not correct for the modules to which Lemma 3.1 is applied. For example, over R = K[X_1] take M = (X_1). Then M is a nonzero graded generalized Eulerian holonomic A_1(K)-module with M_n = 0 for n ≤ −1, and H^0(X_1; M) = M/X_1 M ≅ K is concentrated in degree 1, not degree 0. Consequently the base case of the induction in Lemma 3.1 is not justified by the cited fact as stated.","section":"Section 2.9(2), proof of Lemma 3.1 (m = 1 case)"},{"comment":"The proofs of Theorems 1.1–1.3 depend essentially on Theorems 2.6–2.8, quoted from the author's earlier paper [5] and the unreviewed arXiv preprint [6] (arXiv:2307.04473). The manuscript does not reproduce or prove these structural results, and [6] has not undergone peer review. This reliance would be a concern even if Lemma 3.1 were correct, because the central dichotomy of the paper is imported from an unverified source. As submitted, the verification chain is incomplete at two independent levels.","section":"Theorems 2.6–2.8 and reliance on [6]"}],"minor_comments":[{"comment":"The title contains a spacing error: 'POL YNOMIAL' should be 'POLYNOMIAL'.","section":"Title and abstract"},{"comment":"The heading reads 'with hyotheses as above' and should read 'with hypotheses as above'.","section":"Lemma 2.11"},{"comment":"The sentence 'Then N is FR-finite if char K = p > 0 and M is generalized Eulerian and holonomic if char K = 0' should read 'N is generalized Eulerian and holonomic' in the characteristic-zero case.","section":"Proof of Theorem 1.2(2)"},{"comment":"There are several typographical errors: 'isomorphism's' should be 'isomorphisms', and 'ℓ(Nn) = s for all n ∈ N n' contains a stray 'N' at the end.","section":"Section 5"}],"recommendation":"reject","confidential_remarks":"The paper's central lemma (Lemma 3.1) is false as stated, with a concrete counterexample in characteristic zero, and the same flaw affects the m = 1 base case through the misstated citation in 2.9(2). Because the main theorems all rely on this lemma, the manuscript cannot be accepted in its current form. The heavy dependence on the author's unreviewed preprint [6] further weakens the verification chain. This is not a matter of presentation or a local fix; the proof strategy as written collapses, even if the stated theorems might ultimately be true."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick note: this one is in worse shape than the Pith Report suggests. The central engine, Lemma 3.1, is not just missing a justification — it is false.\n\nLemma 3.1 claims that if a graded generalized Eulerian holonomic A_m-module (or graded F-finite module in char p) has no negative-degree components, then it is free over R. That is wrong. The residue field K is holonomic, generalized Eulerian, has K_n = 0 for n < 0, and is not R^s. The maximal ideal m = (X_1,...,X_m) is another example: it is holonomic, generalized Eulerian, vanishes in negative degrees, and is not free. The stress-test example m = (X_1,X_2) over K[X_1,X_2] is actually in the class, not outside it, and it shows precisely why the proof breaks: X_1 is regular on m, the quotient m/X_1 m is free over K[X_2], yet m is not free over K[X_1,X_2].\n\nThis is a load-bearing flaw. Theorem 1.2 uses Lemma 3.1 to rule out torsion modules that vanish in low degrees, and Theorem 1.1(3) uses the same lemma. Worse, the main theorem as stated has an immediate counterexample. Take R = K[X_1,X_2] and I = (X_1). Then H^0_I(R) = K[X_2] is a graded Lyubeznik functor, it is I-torsion, it is nonzero, its support is not the maximal ideal, and its negative-degree components are all zero. So Theorem 1.2(1) and (2) fail. The paper's own definitions and hypotheses allow this case.\n\nThe paper is clearly written and the high-level program — proving structured nonvanishing and infinite-dimensionality statements for graded components — is reasonable. The background results quoted from the author's earlier work may well be fine. But the new step that is supposed to do the work is false, and the main theorems are false as stated.\n\nMy recommendation: do not send this version to peer review. A referee would bounce it quickly. The author could repair it by adding hypotheses that rule out H^0 and the counterexamples to Lemma 3.1, but that is a substantial rewrite, not a typo fix. If a corrected version appears, it might be worth another look; the current one should not be engaged further.","headline":"The main theorem is false as stated: the key lemma has a counterexample inside its own module class, and H^0_I(R) directly violates Theorem 1.2.","tokens_in":8219,"tokens_out":13039,"would_cite":false,"duration_ms":137794,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D45","14B15","13N10","32C36"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a polynomial ring in $m$ variables, any nonzero local cohomology module of a homogeneous ideal of intermediate height has nonzero graded pieces in every degree $n \\le -m$, and outside the maximal-ideal case it has nonzero pieces in…","keywords":["local cohomology","graded local cohomology","Lyubeznik functors","Koszul homology","ring of differential operators","F-finite modules","holonomic D-modules","graded components"],"falsifier":"Compute the graded components of $H^{m-1}_I(R)$ for $I=(X_1,\\dots,X_{m-1})$ in $R=K[X_1,\\dots,X_m]$ with $m \\ge 3$. The theorem predicts every component is nonzero and, in degree $n \\le -m$, infinite-dimensional; a single zero or finite-dimensional component would refute Theorem 1.1.","tokens_in":7279,"feed_emoji":"♾️","tokens_out":10556,"duration_ms":95291,"temperature":0.7,"pith_summary":"The paper proves that nonzero local cohomology modules of polynomial rings are extremely dense: once a homogeneous ideal of intermediate height ($1 \\le g \\le m-1$) gives a nonzero local cohomology module, that module has a nonzero component in every degree $n \\le -m$, and unless its support is exactly the maximal ideal, it has a nonzero component in every integer degree. In characteristic zero, the stronger statement holds that every graded component is infinite-dimensional when the support is not the maximal ideal, and the top-degree module $H^g_I(R)$ has infinite-dimensional components in every degree regardless of support. These conclusions are drawn for graded Lyubeznik functors, a broad class that includes iterated local cohomology and local cohomology with a variable inverted. The upshot is that local cohomology modules are never sparse at the bottom: apart from the special maximal-ideal module, they occupy every graded position.","feed_headline":"Nonzero local cohomology fills every degree below -m","feed_subtitle":"Nonzero local cohomology modules are dense below degree -m, and their top-degree pieces are infinite-dimensional.","key_machinery":"The engine is a set of structural dichotomy theorems (2.6–2.8) for graded $F$-finite modules in characteristic $p$ and graded generalized Eulerian holonomic modules in characteristic $0$: vanishing in one component in a low, high, or middle range forces vanishing or nonvanishing in every component of a whole half-line. Lemma 3.1 sharpens this: a nonzero module of this class with no components in negative degrees must be a free $R$-module, which contradicts torsion. The top-degree statement in Theorem 1.1(3) additionally uses an exact sequence from multiplication by a linear non-zerodivisor on $H^g_I(R)$, a direct-limit argument over $\\operatorname{Ext}^g_R(R/I^n,R)$, and the associated-prime characterization supplied by reference [7].","core_discovery":"Theorem 1.1 states that for $R=K[X_1,\\ldots,X_m]$ with $m \\ge 2$, a homogeneous ideal $I$ of height $g$ with $1 \\le g \\le m-1$, and a degree $i$ for which $H^i_I(R) \\neq 0$, the following hold: $H^i_I(R)_n \\neq 0$ for all $n \\le -m$; if $\\operatorname{Supp} H^i_I(R) \\neq \\{(X_1,\\ldots,X_m)\\}$, then $H^i_I(R)_n \\neq 0$ for all $n \\in \\mathbb{Z}$; and $\\dim_K H^g_I(R)_n$ is infinite for all $n \\in \\mathbb{Z}$. The same nonvanishing statements are proved for graded Lyubeznik functors $\\mathcal{T}$ for which $\\mathcal{T}(R)$ is $J$-torsion for a nonzero graded ideal $J$, and in characteristic zero the non-maximal-support case has infinite-dimensional components in every degree. A single zero component in a low or middle degree therefore forces the module to be the special maximal-ideal module, which is the only place where graded components can be finite-dimensional or sparse.","pith_inferences":["The dichotomy suggests a testable rigidity principle: for graded modules of this class, the Hilbert function is either identically zero outside a finite range or nonzero on a whole half-line; checking this principle for rings beyond polynomial algebras, such as toric rings, is a natural next step.","The proof of Theorem 1.3 uses Bernstein dimension to rule out finite-dimensional components; the same argument should apply to any graded holonomic module in characteristic zero, yielding a general criterion that a holonomic module with a finite-dimensional component must be supported at the maximal ideal.","The bound $-m$ is plausibly optimal: the theorem predicts no gaps at or below $-m$, but does not forbid a zero component at $-m+1$ when the support is non-maximal; constructing such an example would show the cutoff is sharp while remaining consistent with the paper."],"forward_implications":["For any homogeneous ideal of height $g$ with $1 \\le g \\le m-1$, a nonzero module $H^i_I(R)$ cannot have a gap below degree $-m$; finitely many exceptions at the bottom cannot occur.","If $\\operatorname{Supp} H^i_I(R) \\neq \\{\\mathfrak{m}\\}$, then $H^i_I(R)_n \\neq 0$ for every $n \\in \\mathbb{Z}$; a single zero component forces the maximal-ideal support.","In characteristic zero, finiteness of one component $\\mathcal{T}(R)_{n_0}$ for a graded Lyubeznik functor is equivalent to $\\operatorname{Supp} \\mathcal{T}(R) = \\{\\mathfrak{m}\\}$.","The same nonvanishing statements hold for any graded Lyubeznik functor with $\\mathcal{T}(R)$ torsion over a nonzero graded ideal, covering iterated local cohomology and local cohomology with a variable inverted."],"supporting_citations":[{"why":"Supplies the structural dichotomy theorems 2.6–2.8 for graded local cohomology modules that the proofs invoke as the main engine.","marker":"[5]"},{"why":"The preprint that provides the characteristic-$p$ cases of the structural theorems and the F-finite module facts used in Lemma 2.9.","marker":"[6]"},{"why":"Establishes that graded Lyubeznik functors yield holonomic D-modules in characteristic zero.","marker":"[2]"},{"why":"Supplies the F-finite module framework used throughout the characteristic-$p$ arguments.","marker":"[3]"},{"why":"Provides the generalized Eulerian property and the Koszul-homology facts used in Lemma 2.9.","marker":"[4]"},{"why":"Provides the equality of associated primes, $\\operatorname{Ass} \\operatorname{Ext}^g_R(R/I^n,R) = \\operatorname{Ass} H^g_I(R)$, used in the final contradiction of Theorem 1.1(3).","marker":"[7]"}],"fun_headline_variants":["Nonzero local cohomology fills all degrees below -m","Top local cohomology is infinite in every degree","A zero graded component forces maximal-ideal support","Lyubeznik functors inherit dense nonvanishing below -m"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument rests on structural theorems, quoted from the author's earlier work, claiming that for the special kinds of graded modules that occur here, a zero in a single low or middle degree forces zeros or nonzeros throughout a whole range; if those theorems fail, the paper's conclusions do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Nonzero local cohomology fills all degrees below -m","Top local cohomology is infinite in every degree","A zero graded component forces maximal-ideal support","Lyubeznik functors inherit dense nonvanishing below -m"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000616,"raw_usage":{"total_tokens":2906,"prompt_tokens":1035,"completion_tokens":1871,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":1804}},"tokens_in":651,"tokens_out":1871,"duration_ms":17772,"temperature":1.0,"reasoning_tokens":1804,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:53:31.189247+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the graded components of $H^{m-1}_I(R)$ for $I=(X_1,\\dots,X_{m-1})$ in $R=K[X_1,\\dots,X_m]$ with $m \\ge 3$. The theorem predicts every component is nonzero and, in degree $n \\le -m$, infinite-dimensional; a single zero or finite-dimensional component would refute Theorem 1.1.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the structural dichotomy theorems 2.6–2.8 for graded local cohomology modules that the proofs invoke as the main engine."},{"cited_title":"Lyubeznik, Finiteness Properties of Local Cohomology Modules (an Application of D-modules to Commutative Algebra), Inv","cited_arxiv_id":null,"evidence_quote":"Establishes that graded Lyubeznik functors yield holonomic D-modules in characteristic zero."},{"cited_title":"Reine Angew","cited_arxiv_id":null,"evidence_quote":"Supplies the F-finite module framework used throughout the characteristic-$p$ arguments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the generalized Eulerian property and the Koszul-homology facts used in Lemma 2.9."},{"cited_title":"Schenzel, On the use of local cohomology in algebra and geometry , in, Six lectures on commutative algebra ( B ellaterra, 1996) , Progr","cited_arxiv_id":null,"evidence_quote":"Provides the equality of associated primes, $\\operatorname{Ass} \\operatorname{Ext}^g_R(R/I^n,R) = \\operatorname{Ass} H^g_I(R)$, used in the final contradiction of Theorem 1.1(3)."}],"review_version":1}