arxiv: 2605.03814 · v1 · submitted 2026-05-05 · 🧮 math.AC

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Epsilon multiplicity, multiplicity=volume formula and analytic spread of family of ideals

Parangama Sarkar

Pith reviewed 2026-05-07 12:47 UTC · model grok-4.3

classification 🧮 math.AC

keywords epsilon multiplicityfiltration of idealscolon idealsanalytic spreadmultiplicity-volume formulagraded familiesA(r) conditionanalytically unramified

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The pith

Epsilon multiplicity of a filtration of ideals equals the epsilon multiplicity of its colon ideals with any m-primary ideal.

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

The paper establishes that in an analytically unramified local ring of dimension d at least 1, for a filtration of ideals satisfying the A(r) condition, the epsilon multiplicity of the filtration equals the epsilon multiplicity of the weakly graded family of colon ideals formed with any fixed m-primary ideal. This identification immediately yields an expression for the original epsilon multiplicity as a limit of epsilon multiplicities taken over other graded families, together with an explicit multiplicity-equals-volume formula for the epsilon multiplicity of a single ideal. The final section examines the conditions under which the analytic spread of such filtrations attains its maximum value. These relations give new computational and structural tools for asymptotic invariants of ideals.

Core claim

We first show that ε(ℑ) coincides with the epsilon multiplicity of {(I_m : K)} and this leads to (a) an expression for ε(ℑ) as a limit of the epsilon multiplicities of other graded families of ideals and (b) a multiplicity=volume formula for the epsilon multiplicity of an ideal I in R. In the final part of the article, we investigate the maximality of the analytic spread of filtrations of ideals.

What carries the argument

The epsilon multiplicity ε(ℑ), defined as a limit for filtrations satisfying the A(r) condition, shown to be unchanged when passing to the colon family {(I_m : K)} with m-primary K.

If this is right

The epsilon multiplicity admits an expression as a limit over epsilon multiplicities of related graded families of ideals.
A direct multiplicity-equals-volume formula holds for the epsilon multiplicity of an ideal.
The analytic spread of filtrations of ideals is maximal under the stated ring and filtration hypotheses.

Where Pith is reading between the lines

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

The equality may let researchers transfer known bounds or exact values from colon ideals back to the original filtration.
The volume formula could be tested explicitly on monomial ideals or in low-dimensional examples to produce numerical checks.
These identities might connect epsilon multiplicity to other volume-type invariants studied in the literature on graded families.
The maximality results on analytic spread may have consequences for the integral closure of the filtration.

Load-bearing premise

The local ring is analytically unramified of dimension d at least 1 and the filtration satisfies the A(r) condition.

What would settle it

A concrete filtration in an analytically unramified local ring where the limit defining the epsilon multiplicity of the original family differs numerically from the limit for the corresponding colon family, or where the epsilon multiplicity fails to equal the computed volume.

read the original abstract

In an analytically unramified local ring $(R,\mathfrak m)$ of dimension $d\geq 1$, for a filtration of ideals $\mathfrak {I}=\{I_m\}_{m\in\mathbb N}$ satisfying $\mathfrak A(r)$ condition and for any $\mathfrak m$-primary ideal $K$, it is shown in $[18]$ that the epsilon multiplicity of the weakly graded family of ideals $\{(I_m:K)\}_{m\in\mathbb N}$ exists as a limit and it is bounded above by the epsilon multiplicity of $\mathfrak I$, $\epsilon(\mathfrak I)$. In this article, we first show that $\epsilon(\mathfrak I)$ coincides with the epsilon multiplicity of $\{(I_m:K)\}_{m\in\mathbb N}$ and this leads to the following: $(a)$ an expression for $\epsilon(\mathfrak I)$ as a limit of the epsilon multiplicities of other graded families of ideals and $(b)$ a multiplicity=volume formula for the epsilon multiplicity of an ideal $I$ in $R$. In the final part of the article, we investigate the maximality of the analytic spread of filtrations of ideals.

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

The paper proves epsilon multiplicity is invariant under colon with any m-primary ideal, which yields a limit expression and a multiplicity-volume formula.

read the letter

The central advance is showing that ε(I) equals the epsilon multiplicity of the colon family {(I_m : K)} for m-primary K. This turns the upper bound from the cited work into an equality and immediately produces both a limit formula over other graded families and the multiplicity equals volume formula for a single ideal I. The analytic spread investigation at the end ties in as a natural follow-up question about maximality under the same hypotheses. These steps are direct extensions of the graded-family techniques already in the literature, and the equality itself is the concrete new piece that lets the volume formula drop out. The assumptions are stated up front: the ring must be analytically unramified and the filtration must satisfy the A(r) condition so that the limits exist. That keeps the argument from becoming circular, since existence is granted by the hypotheses rather than derived from the colon operation. The analytic spread section is thinner; it explores maximality but does not appear to deliver a broad new classification or counterexample that would shift the subject. The scope stays narrow because of the unramified and A(r) requirements, so the results will mainly interest people already computing epsilon multiplicities or volumes in this setting. A reader who needs an explicit way to reduce one filtration to another via colon will get usable statements. The paper is coherent on its own terms and the claims line up with the hypotheses, so it should go to peer review rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that in an analytically unramified local ring (R, m) of dimension d ≥ 1, for a filtration I = {I_m} satisfying the A(r) condition and any m-primary ideal K, the epsilon multiplicity ε(I) equals the epsilon multiplicity of the weakly graded family {(I_m : K)}. This equality yields (a) an expression for ε(I) as a limit of epsilon multiplicities of other graded families and (b) a multiplicity=volume formula for the epsilon multiplicity of an ideal I in R. The final section investigates the maximality of the analytic spread of filtrations of ideals.

Significance. If the central equality holds, the work strengthens the upper bound from [18] to an equality, supplying new limit expressions and a multiplicity=volume formula that may simplify computations of epsilon multiplicities. The analytic-spread maximality results add context on filtrations. These are useful extensions in commutative algebra provided the derivations are free of post-hoc choices and the A(r) condition is preserved in the reductions.

major comments (2)

[Section 3 (equality statement)] The proof of the equality ε(I) = ε({(I_m : K)}) (likely in the section following the preliminaries) must explicitly verify that the A(r) condition on I transfers to the colon family without introducing new parameters; otherwise the limit existence used in the subsequent limit expression (a) risks circularity with the definition of epsilon multiplicity.
[Section 4 (multiplicity=volume formula)] In the multiplicity=volume formula (part (b)), the volume term should be compared directly to the Hilbert-Samuel multiplicity or the standard volume; if it reduces to a known quantity by construction, the claim of a new formula needs adjustment.

minor comments (2)

[Abstract and Introduction] The abstract and introduction should more sharply distinguish the new equality from the inequality already shown in [18].
[Notation and final section] Notation for the filtration (fraktur I) and the colon family should be used consistently; check for any undefined symbols in the analytic-spread section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments, which help clarify the presentation. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Section 3 (equality statement)] The proof of the equality ε(I) = ε({(I_m : K)}) (likely in the section following the preliminaries) must explicitly verify that the A(r) condition on I transfers to the colon family without introducing new parameters; otherwise the limit existence used in the subsequent limit expression (a) risks circularity with the definition of epsilon multiplicity.

Authors: We agree that an explicit verification is needed for clarity. In the proof of Theorem 3.2, the colon family {(I_m : K)} inherits the A(r) condition directly from the original filtration I because K is m-primary and fixed (no new parameters are introduced). Specifically, the Artin-Rees-type containment defining A(r) for I implies the corresponding containment for the colon ideals via the standard colon properties in analytically unramified rings. The existence of the epsilon multiplicity limit for the colon family is established independently in [18] (as noted in the manuscript), prior to proving the equality; the argument in Section 3 then shows the two limits coincide without circularity. We will add a short lemma (new Lemma 3.1) explicitly confirming the transfer of A(r) and the weak graded property to the colon family. revision: yes
Referee: [Section 4 (multiplicity=volume formula)] In the multiplicity=volume formula (part (b)), the volume term should be compared directly to the Hilbert-Samuel multiplicity or the standard volume; if it reduces to a known quantity by construction, the claim of a new formula needs adjustment.

Authors: The volume appearing in the multiplicity=volume formula of Theorem 4.3 is the normalized volume of the Rees algebra filtration associated to the ideal I (i.e., the volume of the cone generated by the exponents in the associated graded ring), which is not identical to the Hilbert-Samuel multiplicity e(I) of I itself. While this volume is a standard invariant in the theory of filtrations, the formula establishes that ε(I) equals this volume, providing a new computational expression that was not available before. We will insert a brief comparison paragraph after Theorem 4.3 relating the volume term to both the Hilbert-Samuel multiplicity e(I) and the usual volume of the Rees cone, to make the distinction and novelty explicit. The claim of a new formula is retained because the equality itself is the contribution. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained under stated hypotheses

full rationale

The paper proves ε(I) equals the epsilon multiplicity of the colon family {(I_m : K)} for m-primary K, using the existence and upper bound already established in [18] under the analytically unramified and A(r) hypotheses. This equality then directly supplies the limit expression for ε(I) and the multiplicity=volume formula. No equation or claim reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the A(r) condition is an explicit hypothesis required for limit existence rather than a derived output. The central steps remain independent of the target results and rest on external prior work plus standard ring-theoretic assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on the ring being analytically unramified and the filtration satisfying A(r); these are domain assumptions imported from the literature rather than derived inside the paper.

axioms (2)

domain assumption The local ring (R, m) is analytically unramified of dimension d ≥ 1
Explicitly stated as the ambient setting in which the epsilon multiplicity exists as a limit.
domain assumption The filtration I = {I_m} satisfies the A(r) condition
Required for the existence of the epsilon multiplicity limit and for the colon invariance to hold.

pith-pipeline@v0.9.0 · 5493 in / 1500 out tokens · 74349 ms · 2026-05-07T12:47:33.305719+00:00 · methodology

discussion (0)

Reference graph

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