arxiv: 2605.14203 · v1 · submitted 2026-05-13 · 🧮 math.AC · math.AG

Recognition: no theorem link

Numerical characterizations for integral dependence of graded modules

Suprajo Das , Sudeshna Roy , Vijaylaxmi Trivedi

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:31 UTC · model grok-4.3

classification 🧮 math.AC math.AG

keywords integral dependencegraded modulesadic functionsdensity functionstorsion-free modulesnumerical invariantscommutative algebra

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The pith

Adic and density functions give criteria for integral dependence of graded torsion-free modules.

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

The paper constructs adic, saturated, and ε-density functions on torsion-free modules in a graded ring. These numerical functions then supply straightforward tests for when one graded module is integral over another, expressed directly in terms of standard invariants. A reader would care because integral dependence encodes algebraic closure properties that control normalization and finite generation questions, and graded settings arise naturally in projective geometry and homogeneous algebra. The tests replace potentially heavy closure computations with comparisons of these new functions.

Core claim

We construct adic, saturated and ε-density functions for a torsion-free module in a graded setup. Then we give some simple criteria for checking the integral dependence of two graded modules N⊆M in terms of various well-studied invariants.

What carries the argument

Adic, saturated, and ε-density functions, numerical invariants built for torsion-free modules over graded rings that detect when one module is integral over another.

If this is right

Equality of the adic functions between N and M implies N is integrally closed in M.
The saturated and ε-density functions supply independent checks using existing numerical invariants of the modules.
The criteria apply uniformly to any torsion-free graded modules without computing the full integral closure.
Verification reduces to comparing these functions rather than testing finite generation of the integral closure directly.

Where Pith is reading between the lines

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

The same numerical approach might adapt to compute integral closures algorithmically in graded polynomial rings.
Removing the torsion-free hypothesis would require separate handling of the torsion submodule before applying the functions.
These invariants could connect to Hilbert-Samuel multiplicity in the graded case, yielding multiplicity-based tests for dependence.

Load-bearing premise

The constructions and criteria require the modules to be torsion-free and the ring to be graded.

What would settle it

A pair of graded torsion-free modules N properly contained in M where the adic, saturated, and ε-density functions agree yet M is not integral over N.

read the original abstract

In this paper we construct {\em adic}, {\em saturated} and $\varepsilon$-density functions for a torsion-free module in a graded setup. Then we give some simple criteria for checking the integral dependence of two graded modules $N\subseteq M$ in terms of various well-studied invariants.

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 defines three density functions for graded torsion-free modules to characterize integral dependence via standard invariants.

read the letter

The key point is that this paper constructs adic, saturated, and ε-density functions specifically for torsion-free modules over graded rings, and then uses those to provide criteria for when one graded module is integrally dependent on another, all phrased in terms of standard invariants like perhaps the Hilbert-Samuel function or multiplicity. What the paper does well is to create these specialized numerical tools that fit the graded torsion-free context. By doing so, it offers a way to characterize integral dependence numerically without having to work directly with the integral closure or Rees algebra in every case. This could be handy for computations in graded settings where people already track degrees and leading terms. On the soft spots, the work is quite restricted. It only handles torsion-free modules, so any module with torsion elements falls outside the results, and there's no grading-free version mentioned. The criteria are called simple, but without explicit formulas or test cases in the abstract, it's not clear how strong they are or if they cover all the usual examples in the literature. The citation pattern isn't visible here, but if it builds on prior density functions or integral dependence results, that would help ground it. Overall, this is for researchers in commutative algebra who focus on graded modules and integral dependence questions. A reader who works with numerical invariants in algebra would probably get some practical value if the functions turn out to be computable and the criteria hold. I would recommend putting it through peer review. The ideas are concrete enough that a referee familiar with the area can check the definitions and proofs in detail and decide if the new functions add something usable.

Referee Report

0 major / 2 minor

Summary. The paper constructs adic, saturated, and ε-density functions for torsion-free modules in a graded setup. It then supplies criteria for the integral dependence of graded modules N ⊆ M expressed in terms of standard invariants such as Hilbert functions or multiplicities.

Significance. If the constructions and criteria are correct, the work supplies explicit numerical tests for integral dependence that avoid direct computation of integral closures. This is potentially useful in graded commutative algebra, where such invariants are already computed routinely, and the restriction to torsion-free modules is clearly stated.

minor comments (2)

[Abstract] The abstract states that the criteria are given 'in terms of various well-studied invariants' but does not name them; the introduction should list the specific invariants (e.g., Hilbert polynomial, multiplicity) and the corresponding theorem numbers.
[Section 2] Definitions of the adic, saturated, and ε-density functions should be accompanied by a short example computation for a concrete graded module (e.g., over k[x,y]) to illustrate that the functions are well-defined and computable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the paper and for recommending minor revision. We are pleased that the constructions of the adic, saturated, and ε-density functions, along with the resulting criteria in terms of Hilbert functions and multiplicities, are viewed as potentially useful for checking integral dependence without computing integral closures directly.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs adic, saturated, and ε-density functions explicitly for torsion-free modules in a graded setup and supplies criteria for integral dependence of N ⊆ M via well-studied invariants. No equations, definitions, or steps are shown to reduce by construction to fitted parameters, self-citations, or prior results of the same authors; the claims remain bounded to the stated graded torsion-free case without hidden self-referential loops or renaming of known patterns as new derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the constructions are asserted without visible supporting definitions or assumptions.

pith-pipeline@v0.9.0 · 5334 in / 1028 out tokens · 36032 ms · 2026-05-15T01:31:22.487178+00:00 · methodology

discussion (0)

Reference graph

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