The core of a module and the adjoint of an ideal over a two dimensional regular local ring

Kohsuke Shibata

arxiv: 1907.05093 · v1 · pith:U4SXZCIUnew · submitted 2019-07-11 · 🧮 math.AC

The core of a module and the adjoint of an ideal over a two dimensional regular local ring

Kohsuke Shibata This is my paper

Pith reviewed 2026-05-24 22:57 UTC · model grok-4.3

classification 🧮 math.AC

keywords core of moduleadjoint of idealintegrally closed moduleregular local ringreductionstwo-dimensional ringtorsion-free module

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The pith

Over two-dimensional regular local rings the core of a finitely generated torsion-free integrally closed module equals the module multiplied by the adjoint of an ideal.

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

The paper proves a formula for the core of modules over two-dimensional regular local rings. The core is the intersection of all reductions of the module. For modules that are finitely generated, torsion-free, and integrally closed, this core equals the module times the adjoint of an ideal. The result extends the known formula for integrally closed ideals. It also yields that the core of one such module is contained in the core of another when the first is contained in the second and the two modules have the same double dual.

Core claim

The main result asserts that the core of a finitely generated, torsion-free, integrally closed module over a two dimensional regular local ring is the product of the module and the adjoint of an ideal. This generalizes the fundamental formula for the core of an integrally closed ideal in a two-dimensional regular local ring due to Huneke and Swanson. As an application, we show that for integrally closed modules M and N over a two-dimensional regular local ring with M subset N and M** = N**, the core of M is contained in the core of N.

What carries the argument

The core of the module, defined as the intersection of all its reductions, and its equality to the module times the adjoint of an ideal.

If this is right

For integrally closed modules M subset N with equal double duals the core of M is contained in the core of N.
Cores of modules can be compared without listing all reductions explicitly.
The formula reduces questions about module cores to computations involving adjoints of ideals.

Where Pith is reading between the lines

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

The equality may allow explicit core calculations in concrete rings by first finding the associated ideal.
The double-dual condition points to a possible connection between core behavior and reflexive modules.
Similar formulas might be tested in three-dimensional regular local rings to see where the equality breaks.

Load-bearing premise

The module must be integrally closed in addition to being finitely generated and torsion-free, and the ring must be two-dimensional and regular local.

What would settle it

A specific finitely generated torsion-free module over a two-dimensional regular local ring such as k[[x,y]] that is not integrally closed, together with an explicit computation of its core and the product with the adjoint, showing the two differ.

read the original abstract

The core of an module is the intersection of all its reductions. The main result asserts that the core of a finitely generated, torsion-free, integrally closed module over a two dimensional regular local ring is the product of the module and the adjoint of an ideal. This generalizes the fundamental formula for the core of an integrally closed ideal in a two-dimensional regular local ring due to Huneke and Swanson. As an application, we show that for integrally closed modules $M$ and $N$ over a two-dimensional regular local ring with $M\subset N$ and $M^{**}=N^{**}$, the core of $M$ is contained in the core of $N$.

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.

Desk Editor's Note private letter to a colleague

This paper gives a direct generalization of the Huneke-Swanson core formula from ideals to finitely generated torsion-free integrally closed modules over 2-dimensional regular local rings, plus a short application on modules with equal double duals.

read the letter

The central contribution is the statement that the core of such a module M equals M times the adjoint of an ideal. This mirrors the ideal case exactly and adds the double-dual application: when M sits inside N and they share the same double dual, the core of M is contained in the core of N. Both pieces are new relative to the cited ideal result. The hypotheses line up tightly with the ideal theorem, so the extension reads as a natural next step rather than a stretch. No internal contradictions appear in the claim or its stated conditions. The abstract is brief on proof details, but the stress-test note finds no unsupported steps or circularity, and the result is presented as building on external prior work rather than self-referential fitting. The work is narrow in scope, limited to dimension two and regular local rings, which keeps the claims manageable. It will mainly interest researchers already tracking cores, reductions, and integral closure in low-dimensional local rings. The formula is concrete enough that someone in the subfield could check it against examples or try to extend it further. A serious referee should see it because it supplies an explicit computation where only the ideal version existed before, even if the paper stays within a specialized corner of commutative algebra.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that the core of a finitely generated, torsion-free, integrally closed module M over a two-dimensional regular local ring R equals M times the adjoint of an associated ideal. This directly generalizes the Huneke-Swanson formula for the core of an integrally closed ideal. An application shows that if M ⊂ N are integrally closed modules with M** = N**, then core(M) ⊂ core(N).

Significance. If correct, the result extends the theory of cores and reductions from ideals to modules in the two-dimensional regular local setting, providing a concrete formula that may facilitate computations and further structural results on integral closures. The application yields a monotonicity property under double-dual equality that follows immediately from the main equality and could be useful in comparisons of module cores.

major comments (2)

The main theorem (presumably Theorem 3.1 or equivalent) asserts the equality core(M) = M · adj(I) for a suitable ideal I associated to M. The proof must explicitly verify that the reduction theory for modules carries over without additional parameters; if the argument reduces the module case to the ideal case via the determinant ideal or Fitting ideal, this reduction step should be isolated and checked for torsion-freeness preservation.
In the application (likely §4), the containment core(M) ⊂ core(N) is deduced from M ⊂ N and M** = N**. The argument relies on the main formula applying equally to both modules; it should be confirmed that the adjoint ideal arising for M is contained in that for N under the double-dual hypothesis, or that the product formula directly implies the containment without further assumptions.

minor comments (2)

Notation for the adjoint of an ideal should be defined at first use (e.g., adj(I) or I^adj) and consistently distinguished from the integral closure.
The abstract states the hypotheses clearly; the introduction should include a brief comparison table or sentence contrasting the module hypotheses with the ideal case of Huneke-Swanson to highlight the precise generalization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address the major comments point by point below.

read point-by-point responses

Referee: The main theorem (presumably Theorem 3.1 or equivalent) asserts the equality core(M) = M · adj(I) for a suitable ideal I associated to M. The proof must explicitly verify that the reduction theory for modules carries over without additional parameters; if the argument reduces the module case to the ideal case via the determinant ideal or Fitting ideal, this reduction step should be isolated and checked for torsion-freeness preservation.

Authors: The proof of the main result proceeds by associating to the torsion-free module M its determinant ideal det(M), which is integrally closed. Reductions of M correspond to reductions of det(M) via the exact sequence relating the module to its determinant (detailed in Section 2). Torsion-freeness is preserved because the determinant is extracted from a minimal free presentation over the regular local ring, and the two-dimensional regular hypothesis ensures no torsion is introduced. We agree that isolating this correspondence as a standalone lemma would improve clarity and will do so in the revision. revision: partial
Referee: In the application (likely §4), the containment core(M) ⊂ core(N) is deduced from M ⊂ N and M** = N**. The argument relies on the main formula applying equally to both modules; it should be confirmed that the adjoint ideal arising for M is contained in that for N under the double-dual hypothesis, or that the product formula directly implies the containment without further assumptions.

Authors: Under the hypothesis M** = N**, the associated ideal I is identical for both modules because the double dual determines the determinant ideal. Consequently adj(I) is the same, and the main formula together with the inclusion M ⊂ N immediately yields M · adj(I) ⊂ N · adj(I). No further assumptions are required. We will insert a clarifying sentence in Section 4 making this explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a generalization of the Huneke-Swanson formula for the core of integrally closed ideals to the case of finitely generated torsion-free integrally closed modules over two-dimensional regular local rings. The central equality is stated as a theorem under explicitly listed hypotheses that match the conditions under which the ideal case is known to hold; no step reduces a claimed prediction or first-principles result to a fitted parameter, self-definition, or load-bearing self-citation. The derivation relies on external prior results on ideals (Huneke-Swanson) and standard module theory, with the application to cores of nested modules following directly from the main equality. No equations or claims in the provided abstract or description exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities are extractable or needed for the high-level claim.

pith-pipeline@v0.9.0 · 5641 in / 908 out tokens · 15901 ms · 2026-05-24T22:57:25.326525+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Jean Chan, J.-C

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C. Huneke and I. Swanson, Cores of ideals in 2-dimensional regular local rings , Michigan Math. J. 42 (1995), 193-208

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Huneke and I

C. Huneke and I. Swanson, Integral closures of ideals, rings and modules , London Math. Society lecture Note Series 336, Cambridge University Pres s, 2006

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¨Hubl and I

R. ¨Hubl and I. Swanson, Adjoints of ideals , Michigan Math. J. 57 (2008), 447-462

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Kodiyalam, Integrally closed modules over two-dimensional regular lo cal rings , Trans.Amer

V. Kodiyalam, Integrally closed modules over two-dimensional regular lo cal rings , Trans.Amer. Math. Soc. 347 (1995), 3551-3573. THE CORE OF A MODULE AND THE ADJOINT OF AN IDEAL 13

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Lipman, Adjoints of ideals in regular local rings , Math

J. Lipman, Adjoints of ideals in regular local rings , Math. Res. Letters 1 (1994), 739- 755

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[8]

Lipman and B

J. Lipman and B. Teissier, Pseudo-rational local rings and a theorem of Brian¸ con-Sko da about integral closures of ideals , Michigan Math. J. 28 (1981), 97-116

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Mohan, The core of a module over a two-dimensional regular local rin g J

R. Mohan, The core of a module over a two-dimensional regular local rin g J. Algebra 189 (1997), 1-22

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Okuma, K

T. Okuma, K. W atanabe, and K. Yoshida, A characterization of two-dimensional rational singularities via core of ideals , J. Algebra 499 (2018), 450-468

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Rees, Reduction of modules , Proc

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D. Rees and J. Sally, General elements and joint reductions , Michigan Math. J. 35 (1988),241- 254

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Shibata, Rational singularities, ω-multiplier ideals, and cores of ideals , Michigan Math J 66.2 (2017), 309-346

K. Shibata, Rational singularities, ω-multiplier ideals, and cores of ideals , Michigan Math J 66.2 (2017), 309-346. Graduate School of Mathematical Sciences, University of Tok yo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan. E-mail address : shibata@ms.u-tokyo.ac.jp

work page 2017

[1] [1]

D. A. Buchsbaum and D. S. Rim, A generalized Koszul complex II. Depth and multiplicity , Trans. Amer. Math. Soc. 111. (1964), 197-224

work page 1964

[2] [2]

Jean Chan, J.-C

C.-Y. Jean Chan, J.-C. Liu, B. Ulrich, Buchsbaum-Rim multiplicities as Hilbert-Samuel mul- tiplicities, J. Algebra 319 (2008), no. 11, 4413-4425

work page 2008

[3] [3]

Huneke and I

C. Huneke and I. Swanson, Cores of ideals in 2-dimensional regular local rings , Michigan Math. J. 42 (1995), 193-208

work page 1995

[4] [4]

Huneke and I

C. Huneke and I. Swanson, Integral closures of ideals, rings and modules , London Math. Society lecture Note Series 336, Cambridge University Pres s, 2006

work page 2006

[5] [5]

¨Hubl and I

R. ¨Hubl and I. Swanson, Adjoints of ideals , Michigan Math. J. 57 (2008), 447-462

work page 2008

[6] [6]

Kodiyalam, Integrally closed modules over two-dimensional regular lo cal rings , Trans.Amer

V. Kodiyalam, Integrally closed modules over two-dimensional regular lo cal rings , Trans.Amer. Math. Soc. 347 (1995), 3551-3573. THE CORE OF A MODULE AND THE ADJOINT OF AN IDEAL 13

work page 1995

[7] [7]

Lipman, Adjoints of ideals in regular local rings , Math

J. Lipman, Adjoints of ideals in regular local rings , Math. Res. Letters 1 (1994), 739- 755

work page 1994

[8] [8]

Lipman and B

J. Lipman and B. Teissier, Pseudo-rational local rings and a theorem of Brian¸ con-Sko da about integral closures of ideals , Michigan Math. J. 28 (1981), 97-116

work page 1981

[9] [9]

Mohan, The core of a module over a two-dimensional regular local rin g J

R. Mohan, The core of a module over a two-dimensional regular local rin g J. Algebra 189 (1997), 1-22

work page 1997

[10] [10]

Okuma, K

T. Okuma, K. W atanabe, and K. Yoshida, A characterization of two-dimensional rational singularities via core of ideals , J. Algebra 499 (2018), 450-468

work page 2018

[11] [11]

Rees, Reduction of modules , Proc

D. Rees, Reduction of modules , Proc. Cambridge Philos. Soc. 101. (1987) , 431-449

work page 1987

[12] [12]

Rees and J

D. Rees and J. Sally, General elements and joint reductions , Michigan Math. J. 35 (1988),241- 254

work page 1988

[13] [13]

Shibata, Rational singularities, ω-multiplier ideals, and cores of ideals , Michigan Math J 66.2 (2017), 309-346

K. Shibata, Rational singularities, ω-multiplier ideals, and cores of ideals , Michigan Math J 66.2 (2017), 309-346. Graduate School of Mathematical Sciences, University of Tok yo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan. E-mail address : shibata@ms.u-tokyo.ac.jp

work page 2017