pith. sign in

arxiv: 2605.29544 · v1 · pith:AMXD2O7Pnew · submitted 2026-05-28 · 🧮 math.AC

A sharp bound for the Frobenius test exponents in generalized Cohen-Macaulay local rings

Duong Thi Huong , Pham Hung Quy This is my paper

Pith reviewed 2026-06-29 00:04 UTC · model grok-4.3

classification 🧮 math.AC
keywords generalized Cohen-Macaulay ringFrobenius test exponentlocal cohomologyprime characteristicHartshorne-Speiser-Lyubeznik numberparameter idealFrobenius action
0
0 comments X

The pith

Generalized Cohen-Macaulay local rings of prime characteristic satisfy Fte(R) ≤ ⌈log_p(2n0)⌉ + HSL(R).

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

The paper proves an explicit upper bound on the Frobenius test exponent Fte(R) of parameter ideals in a generalized Cohen-Macaulay local ring (R, m) of prime characteristic p. The bound is stated in terms of n0, the integer such that m to the power n0 annihilates every local cohomology module H^i_m(R) for i less than dim(R), together with the Hartshorne-Speiser-Lyubeznik number HSL(R). A reader would care because the generalized Cohen-Macaulay hypothesis supplies a uniform control on the Frobenius action that turns the test exponent into a computable quantity from these two invariants. The authors present the bound as sharp.

Core claim

Let (R, m) be a generalized Cohen-Macaulay local ring of prime characteristic p. Then Fte(R) ≤ ⌈log_p(2n0)⌉ + HSL(R), where n0 is the integer such that m^{n0} H^i_m(R) = 0 for all i < dim(R).

What carries the argument

The generalized Cohen-Macaulay condition that supplies a uniform power n0 annihilating all lower local cohomology modules, which the authors use to relate the Frobenius action on parameter ideals to the HSL number.

If this is right

  • The Frobenius test exponent of every parameter ideal is finite.
  • The size of Fte(R) is controlled by the annihilator exponent of local cohomology rather than by the full ring structure.
  • The bound grows at most logarithmically with n0.
  • When n0 is small the bound reduces essentially to a multiple of HSL(R).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same style of argument might produce bounds in rings where local cohomology is annihilated by some other fixed ideal instead of a power of m.
  • Explicit values of n0 and HSL(R) in concrete examples would immediately give numerical upper limits on Fte(R).
  • The result suggests testing whether the same combination of log and HSL terms appears in bounds for non-parameter ideals.

Load-bearing premise

The ring must be generalized Cohen-Macaulay so that some power of the maximal ideal annihilates the local cohomology modules in degrees below the dimension.

What would settle it

A single generalized Cohen-Macaulay local ring (R, m) of characteristic p together with its n0 and HSL(R) for which the Frobenius test exponent of some parameter ideal exceeds ⌈log_p(2n0)⌉ + HSL(R).

read the original abstract

Let $(R,\frak m)$ be a generalized Cohen-Macaulay local ring of prime characteristic $p$. In this paper we give a sharp bound for the Frobenius test exponent of parameter ideals. Namely, we prove that $$\mathrm{Fte}(R) \le \lceil \log_p(2n_0)\rceil + \mathrm{HSL}(R),$$ where $n_0$ is the integer such that $\frak m^{n_0} \, H^i_{\frak m}(R) = 0$ for all $i < \mathrm{dim}(R)$, and $\lceil x\rceil$ is the smallest integer that is greater than or equal to $x$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves that if (R, m) is a generalized Cohen-Macaulay local ring of prime characteristic p, then the Frobenius test exponent of parameter ideals satisfies Fte(R) ≤ ⌈log_p(2n0)⌉ + HSL(R), where n0 is the smallest integer such that m^{n0} annihilates H^i_m(R) for all i < dim(R).

Significance. If the result holds, the explicit bound combines the generalized Cohen-Macaulay annihilator exponent n0 with the Hartshorne-Speiser-Lyubeznik number in a sharp way; this supplies a concrete, computable estimate for Fte(R) that is expressed solely in terms of standard invariants of the ring and the Frobenius action.

minor comments (2)
  1. [Introduction] §1 (Introduction): the definitions of Fte(R) and HSL(R) are used in the statement of the main theorem but are not recalled until later sections; a brief reminder in the introduction would improve readability.
  2. [Abstract] The claim that the bound is sharp requires an explicit example (or family of examples) attaining equality; the section containing this verification should be referenced in the abstract or introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive recommendation of minor revision. The referee report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states and proves an explicit upper bound Fte(R) ≤ ⌈log_p(2n0)⌉ + HSL(R) that holds under the generalized Cohen-Macaulay hypothesis (existence of finite n0 annihilating lower local cohomology). n0 and HSL(R) are independently defined ring invariants; the bound is presented as a derived inequality rather than a tautology, fitted parameter, or self-referential definition. No quoted equations or steps reduce the claimed result to its inputs by construction, and no load-bearing self-citations or imported uniqueness theorems appear in the abstract or stated claim. The result is therefore self-contained against external benchmarks in commutative algebra.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard definition of generalized Cohen-Macaulay rings and on the known finiteness of HSL(R) in positive characteristic; no new entities or fitted constants are introduced in the abstract.

axioms (2)
  • domain assumption Existence of finite n0 such that m^{n0} H^i_m(R) = 0 for i < dim(R)
    This is the defining property of generalized Cohen-Macaulay rings invoked in the statement of the bound.
  • domain assumption HSL(R) is finite for generalized Cohen-Macaulay rings of prime characteristic
    The abstract treats HSL(R) as a well-defined finite integer that can be added to the logarithmic term.

pith-pipeline@v0.9.1-grok · 5652 in / 1470 out tokens · 25764 ms · 2026-06-29T00:04:45.730292+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 2 canonical work pages · 1 internal anchor

  1. [1]

    Brenner, Bounds for test exponents, Compos

    H. Brenner, Bounds for test exponents, Compos. Math. 142 (2006), 451--463

    2006

  2. [2]

    Brodmann and R.Y

    M. Brodmann and R.Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Univ. Press, 1998

    1998

  3. [3]

    Cuong, N.T

    N.T. Cuong, N.T. Hoa, and N.T.H. Loan, On certain length functions associated to a system of parameters in local rings, Vietnam J. Math. 27, No. 3 (1999), 259--272

    1999

  4. [4]

    Cuong, and N.T.H

    N.Tu. Cuong, and N.T.H. Loan,

  5. [5]

    Cuong, and L.T

    N.T. Cuong, and L.T. Nhan, Pseudo Cohen-Macaulay and pseudo generalized Cohen-Macaulay modules, J. Algebra 267, No. 1 (2003), 156--177

    2003

  6. [6]

    Cuong, P

    N.T. Cuong, P. Schenzel and N.V. Trung, Verallgemeinerte Cohen-Macaulay moduln, Math. Nachr. 85 (1978), 57--73

    1978

  7. [7]

    Cuong and P.H

    N.T. Cuong and P.H. Quy, A splitting theorem for local cohomology and its applications, J. Algebra 331 (2011), 512--522

    2011

  8. [8]

    Enescu and M

    F. Enescu and M. Hochster, The Frobenius structure of local cohomology, Algebra & Number Theory 2 no. 7 (2008), 721--754

    2008

  9. [9]

    Algebra 85 (1983), pp

    Shiro Goto, On the associated graded rings of parameter ideals in Buchsbaum rings, J. Algebra 85 (1983), pp. 490--534

    1983

  10. [10]

    Gabber, Notes on some t-structures, Geometric aspects of Dwork theory

    O. Gabber, Notes on some t-structures, Geometric aspects of Dwork theory. Vol. II. Berlin: Walter de Gruyter, 2004, 711--734

    2004

  11. [11]

    K. Goel, K. Maddox, L.E. Miller, P.H. Quy, and A. Simpson, A Buchsbaum theory for Frobenius closure, arXiv:2602.03947

    work page arXiv

  12. [12]

    Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans

    M. Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), 463--488

    1977

  13. [13]

    Hochster and C

    M. Hochster and C. Huneke, F -regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 no. 1 (1994), 1--62

    1994

  14. [14]

    Huneke, M

    C. Huneke, M. Katzman, R.Y. Sharp and Y. Yao, Frobenius test exponents for parameter ideals in generalized Cohen-Macaulay local rings, J. Algebra 305 (2006), 516--539

    2006

  15. [15]

    Huong and P.H

    D.T. Huong and P.H. Quy, Notes on the Frobenius test exponents, Comm. Algebra 47 (2019), No. 7, 2702--2710

    2019

  16. [16]

    Huong and P.H

    D.T. Huong and P.H. Quy, Upper bound of multiplicity in prime characteristic, Forum Math. 32 (2020), No. 2, 393--398

    2020

  17. [17]

    Huong and P.H

    D.T. Huong and P.H. Quy, Frobenius test exponent for ideals generated by filter regular sequences, Acta Math. Vietnam. 47 (2022), 151--159

    2022

  18. [18]

    Huong and P.H

    D.T. Huong and P.H. Quy, On the Frobenius closure of parameter ideals in local rings F -injective on the punctured spectrum, Comm. Algebra 51 (2023), No. 11, 4722--4730

    2023

  19. [19]

    Hartshorne and R

    R. Hartshorne and R. Speiser, Local cohomological dimension in characteristic p, Ann. of Math. 105 (1977), 45--79

    1977

  20. [20]

    Katzman and R.Y

    M. Katzman and R.Y. Sharp, Uniform behaviour of the Frobenius closures of ideals generated by regular sequences, J. Algebra 295 (2006), 231--246

    2006

  21. [21]

    Lyubeznik, F -modules: applications to local cohomology and D -modules in characteristic p>0 , J

    G. Lyubeznik, F -modules: applications to local cohomology and D -modules in characteristic p>0 , J. reine angew. Math. 491 (1997), 65--130

    1997

  22. [22]

    Ma, F-injectivity and Buchsbaum singularities, Math

    L. Ma, F-injectivity and Buchsbaum singularities, Math. Ann. 362 (2015), 25--42

    2015

  23. [23]

    Ma and P.H

    L. Ma and P.H. Quy, A Buchsbaum theory for tight closure, Trans. Amer. Math. Soc. 375 (2022), 8257--8276

    2022

  24. [24]

    Maddox, A sufficient condition for finiteness of Frobenius test exponents, Proc

    K. Maddox, A sufficient condition for finiteness of Frobenius test exponents, Proc. Amer. Math.Soc. 147 (2019), no. 12, 5083--5092

    2019

  25. [25]

    Nagel and P

    U. Nagel and P. Schenzel, Cohomological annihilators and Castelnuovo-Mumford regularity, in Commutative algebra: Syzygies, multiplicities, and birational algebra , Contemp. Math. 159 (1994), Amer. Math. Soc. Providence, R.I., 307--328

    1994

  26. [26]

    On the upper semi-continuity of HSL numbers

    S. Murru, On the upper semi-continuity of HSL numbers, preprint, arXiv:1302.1124

    work page internal anchor Pith review Pith/arXiv arXiv

  27. [27]

    Polstra and P.H

    T. Polstra and P.H. Quy, Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals, J. Algebra 529 (2019), 196--225

    2019

  28. [28]

    Quy, A remark on the finiteness dimension, Comm

    P.H. Quy, A remark on the finiteness dimension, Comm. Algebra 41 (2013), 2048--2054

    2013

  29. [29]

    Quy and K

    P.H. Quy and K. Shimomoto, F -injectivity and Frobenius closure of ideals in Noetherian rings of characteristic p> 0 , Adv. Math. 313 (2017), 127--166

    2017

  30. [30]

    Quy, On the uniform bound of Frobenius test exponents, J

    P.H. Quy, On the uniform bound of Frobenius test exponents, J. Algebra 518 (2019), 119--128

    2019

  31. [31]

    Sharp, On the Hartshorne-Speiser-Lyubeznik theorem about Artinian modules with a Frobenius action, Proc

    R.Y. Sharp, On the Hartshorne-Speiser-Lyubeznik theorem about Artinian modules with a Frobenius action, Proc. Amer. Math. Soc. 135 (2007), 665--670

    2007

  32. [32]

    An interaction between algebra, geometry and topology, Springer-Verlag, Berlin, 1986

    Jürgen Stückrad and Wolfgang Vogel, Buchsbaum rings and applications. An interaction between algebra, geometry and topology, Springer-Verlag, Berlin, 1986

    1986

  33. [33]

    Trung, Toward a theory of generalized Cohen--Macaulay modules, Nagoya Math

    N.V. Trung, Toward a theory of generalized Cohen--Macaulay modules, Nagoya Math. J. 102 (1986), 1--49

    1986