A criterion of Cohen Macaulayness of the form module

M. Azeem Khadam

REVIEW 1 major objections 6 minor 13 references

A criterion of Cohen Macaulayness of the form module

T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read One nonzero cohomology level detects Cohen-Macaulayness of form modules.

desk verdict 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. read the letter →

arxiv 1908.07317 v1 pith:FZMO3PKC submitted 2019-08-20 math.AC

M. Azeem Khadam This is my paper · ORCID

classification math.AC MSC 13D4513H10

keywords localcohomologyCohen-MacaulaymoduleformgradeassociatedgradedringČechcomplexdepthRees

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $V(\mathfrak{a}A)$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

Where 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].

In 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.

Who 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.

Referee Report

1 major / 6 minor

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.

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 (1)

[§4.8 (proof of Corollary 4.8), last sentence] 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.

minor comments (6)

[§1, Introduction] 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}".
[§4.6, proof of Theorem 4.6] 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.
[§4.8, proof of Corollary 4.8] 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.
[Acknowledgments] The name "Mateusz Micha/suppress lek" appears garbled and should be corrected to the intended name.
[Example 4.7] 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.
[Notation 2.1(D)] 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.

Circularity Check

1 steps flagged · score 4.0 of 10

Main theorem is independently proven, but the Cohen-Macaulay criterion's converse imports top nonvanishing from the authors' companion paper [7, Theorem 9.5].

self citation load bearing [Corollary 4.8 (proof), used in Corollary 4.9 / Corollary 1.3]
"Note that if a⋆1,...,a⋆t is a system of parameters of GM(q), then a1,...,at is a system of parameters of M. Therefore, ˇLi(a,q,M;n), for some n, is non-zero at i=t and zero afterwards, see [7, Theorem 9.5]."

The converse direction of the advertised Cohen-Macaulay criterion needs the assertion that L^t(a,q,M;n) is nonzero when a is a system of parameters. Without this top nonvanishing, 'exactly one nonzero index' could mean a unique nonzero index less than dim M, so depth < dim and GM(q) would not be Cohen-Macaulay. The paper does not prove this nonvanishing here; it imports it from [7], a companion paper by the same author and Schenzel. Thus a load-bearing part of the criterion's converse rests on an unverified self-citation, rather than on an argument contained in this manuscript.

full rationale

The core result Theorem 4.6 is not circular: it is proved by induction within Section 4 using Lemmas 4.1, 4.2, 4.5, Corollary 4.3, and standard grade/depth facts, and it is not a restatement of any input. The citation of [6, Lemma 4.2] in the proof of Theorem 4.6 is a minor self-citation; it supplies a regular-element lift, and the main argument does not reduce to that lemma. However, Corollary 4.9, advertised as the paper's criterion, relies on Corollary 4.8's use of [7, Theorem 9.5] to assert nonvanishing at the top index t. Since [7] is a companion paper by the present author and Peter Schenzel, and the theorem is not proved or machine-checked here, this is a load-bearing self-citation. It is not a by-construction reduction of the whole derivation, so the appropriate score is 4 rather than 6 or higher.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central theorem depends on the author's companion papers [6] and [7], plus standard results on graded rings. No free parameters are fitted; the system a and exponents c_i are part of the input data. The variation L^i is not invented in this paper, it is taken from [7]. The main inductive proof uses [6, Lemma 4.2] to arrange b in aA. This is the most fragile external input.

assumptions (5)

domain assumption The complex C^bullet(a,q,M;n) and the cohomologies H^i and L^i are defined and satisfy Remark 3.2 and the long exact sequence in Definition 3.3, as established in [7, Section 6].
The paper relies on the companion paper [7] for the construction and the long exact sequence of the variation L^i; Section 3 only repeats the construction.
domain assumption [6, Lemma 4.2]: for any homogeneous G_M(q)-regular element b^* in a^*G_A(q), one can choose b' in q^d \ q^{d+1} with b'^* = b^* and b' in aA.
Used in the induction step of Theorem 4.6 to ensure b lies in aA, which is required for the injectivity and nilpotence argument.
standard math Valabrega-Valla isomorphism G_{M/bM}(q) is isomorphic to G_M(q)/b^*G_M(q) when b^* is G_M(q)-regular.
Cited as [11, Proposition 2.1]; a standard result in the theory of associated graded modules.
standard math Brodmann-Sharp: a finitely generated module is Cohen-Macaulay if and only if local cohomology with support in the maximal ideal vanishes in all but one degree.
Cited as [2, Corollary 6.2.9]; provides the classical analogue that motivates Corollary 4.9.
domain assumption [7, Theorem 9.5]: if a_1,...,a_t is a system of parameters of M, then L^i(a,q,M;n) is nonzero for some n at i = t and zero for i > t.
Used in Corollary 4.8 to obtain non-vanishing at the top degree i = dim G_M(q).

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Pith. "Pith review of A criterion of Cohen Macaulayness of the form module." pith.science (2026). https://pith.science/paper/FZMO3PKC

@misc{pith2026190807317,
  author       = {Pith},
  title        = {Pith review of: A criterion of Cohen Macaulayness of the form module},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FZMO3PKC}},
  note         = {Machine review of arXiv:1908.07317}
}

abstract

Let $\mathfrak{q}$ be an ideal of a Noetherian local ring $(A,\mathfrak{m})$ and $M$ a non-zero finitely generated $A$-module. We present a criterion of Cohen-Macaulayness of the form module $G_M(\mathfrak{q})$ in terms of (non-)vanishing of a variation of local cohomology introduced in \cite{KSch}.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [7]

M. A. Khadam, P. Schenzel: About a Variation of Local Cohomology , to appear to J. Commut. Algebra(2019). 1, 2, 3, 6, 7

work page 2019

[2] [1]

M. F. Atiyah, I. G. Macdonald: ‘Introduction to commutative algebra’, Addison-Wesley Pu bl. Co., Reading, 1969. 2

work page 1969

[3] [2]

Brodmann, R

M. Brodmann, R. Sharp: Local Cohomology. An Algebraic Introduction with Geometri c Applications. Cambridge Studies in Advanced Mathematics No. 60. Cambridg e University Press, (1998). 1, 2

work page 1998

[4] [3]

Bruns, J

W. Bruns, J. Herzog: ‘Cohen-Macaulay rings’, Cambridge Stud. in Advanced Math. , Vol. 39, Cambr. Univ. Press, 1993. 3

work page 1993

[5] [4]

G ˆoto, K.-i

S. G ˆoto, K.-i. W atanabe: On graded rings, I , J. Math. Soc. Japan 30 (1978), 179-213. 2

work page 1978

[6] [5]

Grothendieck: ‘Local cohomology’, Notes by R

A. Grothendieck: ‘Local cohomology’, Notes by R. Hartshorne, Lect. Notes in M ath., 20, Springer,

work page

[7] [6]

M. A. Khadam: On Regular Sequences in the Form Module with App lications to Local B´ezout Inequalities , Bull. Iran. Math. Soc. 44 (2018), 763-779. 6

work page 2018

[8] [8]

Matsumura: ‘Commutative Ring Theory’, Cambridge Univ

H. Matsumura: ‘Commutative Ring Theory’, Cambridge Univ. Press, 1986. 2

work page 1986