On the colength of fractional ideals
Pith reviewed 2026-05-24 17:38 UTC · model grok-4.3
The pith
A recursive formula computes the colength of a fractional ideal from the maximal points of its value set, with the recursion depending on the number of minimal primes.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The central claim is that in complete admissible rings, the colength of a fractional ideal equals a recursive expression whose input consists of the maximal points of the value set of the ideal, with the recursion depth or form governed by the number of minimal primes of the ring; when that number is two or three the expression becomes closed.
What carries the argument
The value set of the fractional ideal, whose maximal points feed the recursive formula for colength.
If this is right
- The colength can be read off the maximal value-set points without computing a basis for the quotient module.
- When the ring has exactly two minimal primes a closed formula for the colength is obtained.
- When the ring has exactly three minimal primes a closed formula for the colength is obtained.
- The same value-set data yields the colength for any fractional ideal in the broader class of complete admissible rings.
Where Pith is reading between the lines
- The recursion might admit a pattern that produces closed formulas for four or more minimal primes.
- The same maximal-point data could be used to compute other length invariants attached to the ideal.
- Explicit numerical examples in rings with known value sets would serve as immediate checks of the recursion.
- The method might extend to non-complete rings if a completion argument preserves the maximal points.
Load-bearing premise
The rings are complete admissible rings, so that the value set of any fractional ideal is well-behaved enough for the recursion to hold.
What would settle it
Take a complete admissible ring with four minimal primes and an explicit fractional ideal whose value set is known; compute the actual colength of the quotient by direct basis count and check whether it equals the number given by the recursive formula.
read the original abstract
The main result in this paper is to supply a recursive formula, on the number of minimal primes, for the colength of a fractional ideal in terms of the maximal points of the value set of the ideal itself. The fractional ideals are taken in the class of complete admissible rings, a more general class of rings than those of algebroid curves. For such rings with two or three minimal primes, a closed formula for that colength is provided, so improving results by Barucci, D'Anna and Fr\"oberg.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript supplies a recursive formula, indexed by the number of minimal primes, for the colength of a fractional ideal in a complete admissible ring, expressed solely in terms of the maximal points of the value set of the ideal. Closed-form expressions are derived for the cases of two and three minimal primes. The setting is strictly larger than that of algebroid curves, and the formulas are presented as improvements on results of Barucci, D'Anna and Fröberg.
Significance. If the derivations hold, the work supplies an explicit, value-set-based method for computing colengths in a broader class of rings than previously treated, together with concrete closed formulas for the low-branch cases. The explicit closed forms constitute a verifiable, usable advance over the earlier literature.
minor comments (3)
- The statement of the main recursive formula (presumably in the section following the preliminaries) should include an explicit base case for a single minimal prime, together with a short verification that it recovers the classical length formula for algebroid curves.
- Notation for the value set V(I) and its maximal points should be fixed once at the beginning of §2 and used consistently; several later displays introduce ad-hoc variants that are not cross-referenced.
- The proof that the recursion terminates after finitely many steps (implicit in the claim that the formula is well-defined) would benefit from a one-sentence remark on the Noetherian property of the value semigroup.
Simulated Author's Rebuttal
We thank the referee for the positive summary and recommendation of minor revision. No major comments appear in the report, so we provide no point-by-point responses.
Circularity Check
No significant circularity in derivation chain
full rationale
The paper claims to derive a recursive formula for colength from the maximal points of the value set of a fractional ideal in complete admissible rings, with closed forms for small numbers of minimal primes. No quoted equations or steps reduce the claimed result to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The derivation is presented as proceeding from the value-set data in a more general ring class than algebroid curves, without the target colength appearing in the inputs by construction. This is the normal case of an independent derivation; the abstract and description supply no evidence of circular reduction.
discussion (0)
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