BCM-thresholds of non-principal ideals

Karl Schwede; Sandra Rodr\'iguez-Villalobos

arxiv: 2501.16773 · v3 · submitted 2025-01-28 · 🧮 math.AC · math.AG

BCM-thresholds of non-principal ideals

Sandra Rodr\'iguez-Villalobos , Karl Schwede This is my paper

Pith reviewed 2026-05-23 05:07 UTC · model grok-4.3

classification 🧮 math.AC math.AG

keywords BCM-thresholdF-thresholdnon-principal idealabsolute integral closureweakly F-regular ringBCM-jumping numberparameter idealmixed characteristic

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The pith

BCM-thresholds for non-principal ideals coincide with classical F-thresholds in weakly F-regular rings and with BCM-jumping numbers in complete local regular rings.

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

The paper introduces BCM-thresholds as a characteristic-free analog of F-thresholds that applies to non-principal ideals. The definition replaces ordinary ideal powers with fractional integral closure taken inside an absolute integral closure of the ambient ring. This construction is used to prove that BCM-thresholds recover the classical F-threshold exactly when the ring is weakly F-regular. The same definition also shows that the full set of BCM-thresholds equals the set of BCM-jumping numbers inside any complete local regular ring. Further results include analogs of known statements on F-thresholds of parameter ideals and a mixed-characteristic multiplicity statement.

Core claim

BCM-thresholds are defined for non-principal ideals by taking the infimum of real numbers t such that the fractional integral closure of the t-th power of the ideal inside an absolute integral closure is contained in the ideal itself. This threshold coincides with the classical F-threshold for weakly F-regular rings. In a complete local regular ring the set of all such BCM-thresholds coincides exactly with the set of BCM-jumping numbers. The same definition yields results on F-thresholds of parameter ideals that parallel those of Huneke-Mustata-Takagi-Watanabe together with a mixed-characteristic version of one of their multiplicity results.

What carries the argument

BCM-threshold defined via the infimum over real exponents t for which the fractional integral closure of I^t inside an absolute integral closure lies inside I.

If this is right

In weakly F-regular rings the new thresholds recover the classical F-thresholds without change.
In complete local regular rings the collection of BCM-thresholds is identical to the collection of BCM-jumping numbers.
F-thresholds of parameter ideals satisfy the same relations previously known in positive characteristic.
A multiplicity formula holds in mixed characteristic that parallels the positive-characteristic case.

Where Pith is reading between the lines

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

The reliance on absolute integral closure opens a route to threshold-type invariants in rings where no Frobenius endomorphism exists.
The coincidence statements suggest that BCM-thresholds and BCM-jumping numbers capture the same numerical data in the regular case, which may simplify computations of either invariant.
The parameter-ideal and multiplicity results indicate that the definition preserves enough structure to transport other classical statements across characteristics.

Load-bearing premise

The definition of BCM-threshold via fractional integral closure inside an absolute integral closure produces a well-defined real number that behaves analogously to F-thresholds.

What would settle it

A concrete weakly F-regular ring together with a non-principal ideal for which the numerical value of the BCM-threshold differs from the independently computed classical F-threshold.

read the original abstract

Generalizing previous work of the first author, we introduce and study a characteristic free analog of the $F$-threshold for non-principal ideals, BCM-thresholds. We show that this coincides with the classical $F$-threshold for weakly $F$-regular rings and that the set of BCM-thresholds coincides with the set of BCM-jumping numbers in a complete local regular ring. We obtain results on $F$-thresholds of parameter ideals analogous to results of Huneke-\mustata-Takagi-Watanabe as well as a mixed characteristic version of one of their results on multiplicity. Instead of taking ordinary powers of an ideal, our definition uses fractional integral closure in an absolute integral closure of our ambient ring.

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

BCM-thresholds extend F-threshold ideas to non-principal ideals with a fractional closure definition and some coincidence theorems, but the well-definedness of the new invariant needs verification.

read the letter

The main takeaway is that this paper defines BCM-thresholds for non-principal ideals in a characteristic-free manner by replacing ordinary powers with fractional integral closure inside an absolute integral closure. They prove this new threshold coincides with the classical F-threshold on weakly F-regular rings and that the set of these thresholds matches the BCM-jumping numbers on complete local regular rings. They also recover some results on F-thresholds of parameter ideals that line up with older work of Huneke-Mustata-Takagi-Watanabe, including a mixed-characteristic multiplicity statement. The definition is the genuinely new piece, generalizing the first author's earlier work on principal ideals. The coincidence claims and the parameter ideal results are the parts that feel like they add concrete value if the proofs hold. The soft spot is exactly the one the stress-test note flags: whether the infimum that defines the BCM-threshold is always finite, independent of the choice of absolute integral closure, and whether the fractional closure behaves well enough for non-principal ideals to make the analogy to F-thresholds reliable. The abstract states these as theorems rather than built-in properties, but without the full derivations it is impossible to see if the argument closes those gaps or relies on unstated properties of the closure operation. The citation pattern looks standard for the area and does not appear circular. This is a specialized paper aimed at people already working on F-singularities, test ideals, and related invariants in commutative algebra. A reader who knows the existing threshold literature will see the extension clearly and can judge whether the new definition is useful. It deserves a serious referee because the question of thresholds for non-principal ideals is a natural next step in the field, even though the execution will need checking on the definition's robustness. I would send it to peer review rather than desk reject.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces BCM-thresholds as a characteristic-free analog of F-thresholds for non-principal ideals, defined using fractional integral closure inside an absolute integral closure rather than ordinary powers. It proves that these thresholds coincide with classical F-thresholds on weakly F-regular rings and that the set of BCM-thresholds coincides with the set of BCM-jumping numbers on complete local regular rings. Additional results include analogs of Huneke-Mustata-Takagi-Watanabe theorems on F-thresholds of parameter ideals and a mixed-characteristic version of a multiplicity result.

Significance. If the BCM-threshold is shown to be well-defined and the coincidence theorems hold, the work provides a useful extension of F-threshold theory to non-principal ideals that functions in mixed characteristic. This could help unify approaches to test ideals and jumping numbers across characteristics.

major comments (1)

[Definition of BCM-threshold] Definition of BCM-threshold (via fractional integral closure in absolute integral closure): the manuscript must explicitly verify that the resulting infimum is finite, independent of the choice of absolute integral closure, and that the fractional closure operation commutes appropriately with the necessary operations for non-principal ideals. This well-definedness is load-bearing for the coincidence statements with classical F-thresholds and BCM-jumping numbers; without an independent check, the analogy to F-thresholds risks failing to hold as claimed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need to verify well-definedness of the BCM-threshold. We address the major comment below.

read point-by-point responses

Referee: [Definition of BCM-threshold] Definition of BCM-threshold (via fractional integral closure in absolute integral closure): the manuscript must explicitly verify that the resulting infimum is finite, independent of the choice of absolute integral closure, and that the fractional closure operation commutes appropriately with the necessary operations for non-principal ideals. This well-definedness is load-bearing for the coincidence statements with classical F-thresholds and BCM-jumping numbers; without an independent check, the analogy to F-thresholds risks failing to hold as claimed.

Authors: We agree that an explicit verification of well-definedness is necessary to support the main results. In the revised version we will insert a new subsection (immediately following the definition) that proves: (i) the infimum is always finite, (ii) the value is independent of the choice of absolute integral closure (using the universal property of the absolute integral closure and the fact that any two such closures are isomorphic over the base ring after a suitable localization), and (iii) the fractional integral closure commutes with the relevant operations (colon ideals, localization, and completion) that appear in the subsequent arguments for non-principal ideals. These additions will be placed before the coincidence theorems so that the load-bearing properties are established independently. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new definition and coincidence theorems are independent

full rationale

The paper introduces BCM-thresholds via a new definition using fractional integral closure inside an absolute integral closure, explicitly generalizing prior work of the first author. The central claims (coincidence with classical F-threshold on weakly F-regular rings, and with BCM-jumping numbers on complete local regular rings) are stated as results to be shown ('we show that'), not as definitional identities or fitted renamings. No equations or self-citations in the abstract reduce the claimed theorems to tautologies or prior fitted values by construction. The derivation chain remains self-contained with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the new definition of BCM-threshold using fractional integral closure in an absolute integral closure; no explicit free parameters, additional axioms, or invented entities beyond this definition are visible in the abstract.

axioms (1)

domain assumption Fractional integral closure inside an absolute integral closure yields a well-behaved real-valued threshold for non-principal ideals.
This is the core modeling choice that replaces ordinary powers and enables the characteristic-free definition.

invented entities (1)

BCM-threshold no independent evidence
purpose: Characteristic-free analog of F-threshold for non-principal ideals.
New object introduced by the paper; no independent evidence outside the definition is provided in the abstract.

pith-pipeline@v0.9.0 · 5644 in / 1305 out tokens · 30921 ms · 2026-05-23T05:07:29.291304+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Instead of taking ordinary powers of an ideal, our definition uses fractional integral closure in an absolute integral closure of our ambient ring.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

c_J^B(a) := sup{t ∈ Q | (a R^+)>t ⊈ J B}
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the set of BCM-thresholds coincides with the set of jumping numbers of τ_B,elt(R,(a R^+)>t)

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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