{"id":"40fda041-a00f-4d11-8865-d598396b01e4","arxiv_id":"1908.07317","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The grade of the leading-form ideal in the form ring equals the least index at which the Khadam-Schenzel local cohomology variant L^i does not vanish.","lead":"This paper proves a criterion for when the associated graded module of an ideal is Cohen-Macaulay, phrased in terms of a recently introduced variant of local cohomology. The criterion says the grade of the leading-form ideal equals the first index at which the variant does not vanish.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.6 is likely sound; the cited [6, Lemma 4.2] is elementary, while the unverified [7, Theorem 9.5] is load-bearing for Corollary 4.9's converse.","rationale":"The paper's central theorem 4.6 is a grade formula for initial-form ideals in terms of the variation \\check L^i. The proof is an induction on grade, and every step I traced is internally coherent: Lemma 4.2 implies the strong divisibility condition needed for Corollary 4.3(2); the Valabrega-Valla isomorphism is justified by that same condition; the support argument plus b in aA gives vanishing below grade; and nonvanishing at grade follows from the long exact sequence once the order of the two final paragraphs is swapped. The external lemma [6, Lemma 4.2] is actually elementary: since b^* is a homogeneous element of the ideal generated by the a_i^*, one can choose homogeneous coefficients in q^{d-c_i}, lift them, and sum; this gives b' in aA with b'^* = b^* and b' not in q^{d+1}. Hence the specific concern raised by the reader is not the real vulnerability. The genuine load-bearing dependency is [7, Theorem 9.5], used in Corollary 4.8 to assert top nonvanishing. Corollary 4.9's converse needs that if depth < dim then there is a nonzero \\check L^i at i = dim (or at least at some index above depth), so that a non-CM module cannot have exactly one nonzero index. Without seeing the proof of [7, Theorem 9.5], this direction is not established within the present manuscript. Therefore the CONDITIONAL verdict stands, but the condition should be verification of [7, Theorem 9.5] rather than [6, Lemma 4.2].","tokens_in":8813,"tokens_out":32380,"duration_ms":277144,"concrete_test":"Obtain the companion paper [7] and verify Theorem 9.5 from the definition of \\check L^i in Definition 3.3; in particular, compute \\check L^2(a,q,M;n) for A = k[[x,y]], q = m, M = A, a = (x,y), using only the definitions in Section 3, and check that it is nonzero for some n. If this top nonvanishing fails, Corollary 4.8's assertion fails and the converse of the Cohen-Macaulay criterion in Corollary 4.9 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4.6 invokes [6, Lemma 4.2] to replace a regular element b^* in a^*G_A(q) by b' in aA with the same initial form. This step is actually innocuous: for any homogeneous b^* of degree d in the ideal a^*G_A(q), write b^* = sum f_i a_i^* with f_i homogeneous of degree d-c_i; lift each f_i to an element of q^{d-c_i} and set b' = sum f_i a_i. Then b' is in aA and b'^* = b^* is nonzero, so b' is in q^d \\ q^{d+1}; regularity of b^* is not even needed for the existence of such a lift. Thus the reader's weakest assumption, while cited to [6], is not a real risk. The genuinely load-bearing unverified input is [7, Theorem 9.5], invoked in Corollary 4.8 to assert \\check L^t(a,q,M;n) is nonzero for some n when a is a system of parameters. This top nonvanishing is used in Corollary 4.9 to ensure that if depth < dim, at least two distinct indices have nonzero \\check L^i, so 'exactly one nonzero' implies depth = dim and hence Cohen-Macaulayness. Without a proof of this theorem, the converse direction of the paper's advertised criterion is unestablished; Theorem 4.6 alone identifies only the least nonzero index and does not force nonvanishing at the top.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a criterion for the Cohen-Macaulayness of the form module G_M(q) in terms of the (non-)vanishing of a variation of local cohomology, denoted \\check L^i(a,q,M;n), which was introduced by the author and Schenzel in a companion paper. After collecting preliminaries and defining \\check L^i as the cohomology of a quotient of the \\check Cech complex, the paper proves Theorem 4.6: grade(a^*G_A(q), G_M(q)) is the least integer i for which \\check L^i(a,q,M;n) is nonzero for some n. The proof goes by induction on the grade, using an exact sequence of complexes from Corollary 4.3 and replacing a regular element by a lift in aA. Corollaries 4.8 and 4.9 then assert that, when a^* is a system of parameters, the nonzero indices lie between depth G_M(q) and dim G_M(q), and that G_M(q) is Cohen-Macaulay if and only if \\check L^i(a,q,M;n) is nonzero for exactly one index i.","tokens_in":9071,"tokens_out":12432,"duration_ms":124356,"significance":"If the result holds, Theorem 4.6 gives a clean cohomological formula for the grade of the initial-form ideal in the form ring, and Corollary 4.9 provides a local-cohomology-style criterion for Cohen-Macaulayness of form modules. The core argument for the least-index formula is explicit and is mostly self-contained once one replaces the invocation of [6, Lemma 4.2] by an elementary lifting argument. The main weakness is that the converse direction of the advertised criterion relies on a top-nonvanishing statement taken from the author's companion paper [7, Theorem 9.5], which is not proved or stated in this manuscript; without that input, Corollary 4.9 does not follow from Theorem 4.6 alone.","major_comments":[{"comment":"The proof asserts: \"Therefore, \\check L^i(a,q,M;n), for some n, is non-zero at i = t and zero afterwards, see [7, Theorem 9.5].\" This top nonvanishing is load-bearing for Corollary 4.9's converse. Theorem 4.6 alone only locates the least nonzero index, namely depth G_M(q); it does not force a nonzero at the top index dim G_M(q). Without nonvanishing at i = t, the statement \"exactly one nonzero index\" would not imply depth G_M(q) = dim G_M(q), so the \"only if\" direction of the Cohen-Macaulay criterion would be unproved. Since [7] is a companion paper that is only cited and not reproduced, I request that the full statement of [7, Theorem 9.5] be included, or better, that a proof of this top-nonvanishing be supplied in the present manuscript.","section":"§4.8 (proof of Corollary 4.8), last sentence"}],"minor_comments":[{"comment":"The displayed definition of a^\\star_i reads \"a^\\star_i = a + q^{c_i+1}\"; it should be \"a^\\star_i = a_i + q^{c_i+1}\".","section":"§1, Introduction"},{"comment":"The reliance on [6, Lemma 4.2] is unnecessary and can be replaced by a direct argument: if b^* is a homogeneous element of degree d in a^*G_A(q), write b^* = \\sum f_i a_i^* with f_i homogeneous of degree d-c_i, lift each f_i to an element of q^{d-c_i}, and set b' = \\sum f_i a_i. Then b' \\in aA and (b')^* = b^*, so b' \\in q^d \\setminus q^{d+1}. Adding this one-line argument would make the proof self-contained at this point.","section":"§4.6, proof of Theorem 4.6"},{"comment":"The phrase \"non-zero at i = t and zero afterwards\" should be clarified to \"nonzero for i = t and zero for i > t\", since the intended meaning is not that only the single index i = t is nonzero.","section":"§4.8, proof of Corollary 4.8"},{"comment":"The name \"Mateusz Micha/suppress lek\" appears garbled and should be corrected to the intended name.","section":"Acknowledgments"},{"comment":"The notation \"k[|t^4,t^5,t^11|]\" is nonstandard; it should be \"k[[t^4,t^5,t^11]]\" for consistency with the later use of formal power series.","section":"Example 4.7"},{"comment":"In the two short exact sequences, the shift notation R_M(q)_+[1] is used without defining the shift convention; please state that [1] denotes the standard degree shift by one.","section":"Notation 2.1(D)"}],"recommendation":"major_revision","confidential_remarks":"The central formula is defensible, but the paper's advertised criterion is conditional on [7, Theorem 9.5], which is the author's own companion paper listed as to appear. Since this is load-bearing for the converse of the main criterion, the editor may wish to require that the author provide a self-contained proof or a precise statement with a verifiable proof in an appendix. The manuscript also needs a careful editorial pass for the numerous typographical errors."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a competent short note. The new thing is Theorem 4.6, which expresses grade(a*G_A(q), G_M(q)) as the least i with L^i(a,q,M;n) nonzero for some n. Corollary 4.9 then turns that into a Cohen-Macaulay criterion for G_M(q) when a* is a system of parameters. The result is new relative to the cited literature and does not collapse by definition into an earlier theorem. The main proof is a clean induction on grade, using the short exact sequence of complexes from Corollary 4.3, and the local cohomology setup in Section 3 is standard. Example 4.7 gives a useful sanity check distinguishing the role of q.\n\nWhere I want more: the proof leans on the companion paper [7] for the definition and properties of L^i, and most importantly for Corollary 4.8's assertion that if a1,...,at is a system of parameters of M then L^t(a,q,M;n) is nonzero for some n. That top nonvanishing is what makes \"exactly one nonzero index\" imply depth = dim, i.e., the converse direction of the advertised criterion. The reader flagged [6, Lemma 4.2] as a weak point; I think that concern is minor. The lifting of b* to b' in aA can be done directly by taking homogeneous lifts of the coefficients, so that lemma is not a real source of risk. The genuine load-bearing unverified input is [7, Theorem 9.5].\n\nIn short: Theorem 4.6 itself is likely sound, but the paper as written does not prove the top nonvanishing and therefore the full Cohen-Macaulay criterion is conditional on a theorem in a companion paper that is not reproduced here. That is common in this area, but it should be stated more plainly or the relevant argument sketched.\n\nWho it is for: commutative algebraists working on associated graded modules, Rees algebras, or local cohomology. It is an incremental but real contribution, not a breakthrough. I would send it to a serious referee, mainly to check the invocation of [7, Theorem 9.5] and the induction in Theorem 4.6. I would not desk-reject it.","headline":"Short note with a genuinely new grade formula for form modules; Theorem 4.6 looks sound, but the advertised Cohen-Macaulay criterion depends on a top-nonvanishing result imported from a companion paper.","tokens_in":9657,"tokens_out":1471,"would_cite":true,"duration_ms":14797,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D45","13H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"One nonzero cohomology level detects Cohen-Macaulayness of form modules.","keywords":["local cohomology","Cohen-Macaulay module","form module","grade","associated graded ring","Čech complex","depth","Rees module"],"falsifier":"Compute the two sides of the claimed equality in a concrete case where the lifting lemma has not been checked, such as a monomial ideal in $k[x,y,z]$ or the paper's own example ring $k[[t^4,t^5,t^{11}]]$ with different choices of $\\mathfrak{q}$ and $\\mathfrak{a}$; a mismatch between the grade and the least nonvanishing index would disprove Theorem 4.6.","tokens_in":2016,"feed_emoji":"🧮","tokens_out":7060,"duration_ms":123522,"temperature":0.7,"pith_summary":"This paper proves a cohomological criterion for the form module $G_M(\\mathfrak{q})$ to be Cohen-Macaulay. The criterion is phrased through a variant $\\check{L}^i(\\mathfrak{a},\\mathfrak{q},M;n)$ of local cohomology introduced in a companion paper. The main theorem says that the grade of the initial-form ideal $\\mathfrak{a}^*G_A(\\mathfrak{q})$ in $G_M(\\mathfrak{q})$ is exactly the smallest $i$ for which some $\\check{L}^i$ is nonzero. When $\\mathfrak{a}^*$ is a system of parameters, this gives a Cohen-Macaulay test: the form module is Cohen-Macaulay precisely when $\\check{L}^i$ is nonzero for exactly one index. The result matters because form modules encode the asymptotic behavior of powers of an ideal, and a cohomological test for their Cohen-Macaulayness clarifies when associated graded rings have good properties.","feed_headline":"One cohomology level decides Cohen-Macaulayness of form modules","feed_subtitle":"Form modules are Cohen-Macaulay when exactly one variation-of-local-cohomology level is nonzero.","key_machinery":"The central object is a variation of local cohomology: $\\check{L}^\\bullet(\\mathfrak{a},\\mathfrak{q},M;n)$ is the cohomology of the quotient of the Čech complex of $\\mathfrak{a}$ by a subcomplex built from the $\\mathfrak{q}$-adic filtration, so it refines $H^i_{\\mathfrak{a}A}(M)$ while remembering powers of $\\mathfrak{q}$. The inductive proof chooses a homogeneous regular element $b^*\\in\\mathfrak{a}^*G_A(\\mathfrak{q})$, lifts it to an element $b$ of $\\mathfrak{a}A$ via a cited lemma from the author's earlier work, passes to $G_{M/bM}(\\mathfrak{q})$ using a standard form-ring isomorphism, and then kills the cohomology through the fact that its support lies in $V(\\mathfrak{a}A)$.","core_discovery":"For an ideal $\\mathfrak{q}$ of a Noetherian local ring $A$, a sequence $\\mathfrak{a}=a_1,\\dots,a_t$ with $a_i\\in\\mathfrak{q}^{c_i}$, and a nonzero finite $A$-module $M$, the paper defines a modified Čech complex whose cohomology is $\\check{L}^i(\\mathfrak{a},\\mathfrak{q},M;n)$. Theorem 4.6 asserts that the grade of $\\mathfrak{a}^*G_A(\\mathfrak{q})$ in the form module $G_M(\\mathfrak{q})$ is exactly the least $i$ for which some $\\check{L}^i(\\mathfrak{a},\\mathfrak{q},M;n)$ is nonzero. When $\\mathfrak{a}^*$ is a system of parameters, Corollary 4.9 turns this into the advertised criterion: $G_M(\\mathfrak{q})$ is Cohen-Macaulay over $G_A(\\mathfrak{q})$ if and only if $\\check{L}^i$ is nonzero for exactly one index $i$. The proof inducts on the grade, using the short exact sequences of Corollary 4.3 to pass from $M$ to $M/bM$ when $b^*$ is regular, and uses support containment in $V(\\mathfrak{a}A)$ to force injectivity.","pith_inferences":["Not in the paper: the least-index formula could be read as defining a depth-type invariant for filtered modules even when $\\mathfrak{a}^*$ is not a system of parameters, and it is testable whether this invariant depends only on the integral closure filtration of $\\mathfrak{q}$.","Not in the paper: because $\\check{L}^i(\\mathfrak{a},\\mathfrak{q},M;n)$ is the degree-$n$ component of local cohomology of the Rees module, the theorem may connect asymptotic vanishing in graded local cohomology to Hilbert-coefficient questions.","Not in the paper: in examples where $G_A(\\mathfrak{q})$ has an explicit presentation, the defining complexes can be written down and the least nonzero index computed by a computer algebra system, giving an independent check of the criterion.","Not in the paper: the criterion may suggest analogous Cohen-Macaulay tests for other filtrations, such as integral closures or symbolic powers, whenever the analogous lifting lemma holds."],"forward_implications":["If Theorem 4.6 holds, Cohen-Macaulayness of the form module can be detected by testing the modules $\\check{L}^i(\\mathfrak{a},\\mathfrak{q},M;n)$ rather than by building a full resolution.","In the parameter case, nonvanishing of $\\check{L}^i$ is confined to the interval between $\\operatorname{depth} G_M(\\mathfrak{q})$ and $\\dim G_M(\\mathfrak{q})$, and both endpoints are actually attained.","When $\\mathfrak{q}=\\mathfrak{a}A$, the modules $\\check{L}^i$ agree with ordinary local cohomology for large $n$, so the classical Cohen-Macaulay criterion appears as a special case.","Replacing $\\mathfrak{a}$ by another system with the same radical does not change $\\check{L}^i$, so the criterion is insensitive to the choice of generators for $\\mathfrak{a} A$.","The criterion gives a ready proof that $G_M(\\mathfrak{q})$ is not Cohen-Macaulay whenever more than one index contributes: one needs only one nonzero $\\check{L}^i$ outside the allowed interval."],"supporting_citations":[{"why":"introduces the variation $\\check{L}^i(\\mathfrak{a},\\mathfrak{q},M;n)$ and supplies the long exact sequence and vanishing properties used throughout the proof.","marker":"[7]"},{"why":"provides the cited Lemma 4.2, which lifts a regular initial form $b^*$ to an element $b\\in\\mathfrak{a}A$, a step the induction in Theorem 4.6 depends on.","marker":"[6]"},{"why":"supplies the isomorphism $G_M(\\mathfrak{q})/b^*G_M(\\mathfrak{q})\\cong G_{M/bM}(\\mathfrak{q})$ and the resulting decrease of grade used to continue the induction.","marker":"[11]"},{"why":"is the standard source for grades, maximal regular sequences, and the local cohomology Cohen-Macaulay criterion that the result parallels.","marker":"[2]"},{"why":"contains the explicit computations showing the example ring's form ring is not Cohen-Macaulay, used in Example 4.7 to contrast $\\mathfrak{q}=\\mathfrak{a}A$ with $\\mathfrak{q}=\\mathfrak{m}$.","marker":"[12]"}],"fun_headline_variants":["Cohen-Macaulay form modules: a one-level cohomology test","Exact one nonzero cohomology level characterizes form modules","A single cohomology index decides form module Cohen-Macaulayness","Form module Cohen-Macaulayness: a local cohomology criterion","One nonzero cohomology level marks Cohen-Macaulay form modules"],"cache_read_input_tokens":11648,"weakest_assumption_plain":"The proof depends on a cited lemma of the author's earlier work claiming that every homogeneous regular element of the initial-form ideal can be lifted to an ordinary element of $\\mathfrak{a}A$; if that lifting is impossible for some ideal and module, the induction proving Theorem 4.6 breaks.","fun_headline_variants_meta":{"raw":{"variants":["Cohen-Macaulay form modules: a one-level cohomology test","Exact one nonzero cohomology level characterizes form modules","A single cohomology index decides form module Cohen-Macaulayness","Form module Cohen-Macaulayness: a local cohomology criterion","One nonzero cohomology level marks Cohen-Macaulay form modules"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000177,"raw_usage":{"total_tokens":1249,"prompt_tokens":860,"completion_tokens":389,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":476,"completion_tokens_details":{"reasoning_tokens":293}},"tokens_in":476,"tokens_out":389,"duration_ms":4333,"temperature":1.0,"reasoning_tokens":293,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:21:32.937856+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the two sides of the claimed equality in a concrete case where the lifting lemma has not been checked, such as a monomial ideal in $k[x,y,z]$ or the paper's own example ring $k[[t^4,t^5,t^{11}]]$ with different choices of $\\mathfrak{q}$ and $\\mathfrak{a}$; a mismatch between the grade and the least nonvanishing index would disprove Theorem 4.6.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the variation $\\check{L}^i(\\mathfrak{a},\\mathfrak{q},M;n)$ and supplies the long exact sequence and vanishing properties used throughout the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the cited Lemma 4.2, which lifts a regular initial form $b^*$ to an element $b\\in\\mathfrak{a}A$, a step the induction in Theorem 4.6 depends on."},{"cited_title":"V alabrega, G","cited_arxiv_id":null,"evidence_quote":"supplies the isomorphism $G_M(\\mathfrak{q})/b^*G_M(\\mathfrak{q})\\cong G_{M/bM}(\\mathfrak{q})$ and the resulting decrease of grade used to continue the induction."},{"cited_title":"Brodmann, R","cited_arxiv_id":null,"evidence_quote":"is the standard source for grades, maximal regular sequences, and the local cohomology Cohen-Macaulay criterion that the result parallels."},{"cited_title":"V alla: Hilbert functions of graded algebras , in: J","cited_arxiv_id":null,"evidence_quote":"contains the explicit computations showing the example ring's form ring is not Cohen-Macaulay, used in Example 4.7 to contrast $\\mathfrak{q}=\\mathfrak{a}A$ with $\\mathfrak{q}=\\mathfrak{m}$."}],"review_version":1}