The arithmetic rank of the residual intersections of a complete intersection ideal

Kesavan Mohana Sundaram; Manav Batavia; Taylor Murray; Vaibhav Pandey

arxiv: 2510.17049 · v2 · submitted 2025-10-19 · 🧮 math.AC · math.AG

The arithmetic rank of the residual intersections of a complete intersection ideal

Manav Batavia , Kesavan Mohana Sundaram , Vaibhav Pandey , Taylor Murray This is my paper

Pith reviewed 2026-05-18 06:24 UTC · model grok-4.3

classification 🧮 math.AC math.AG

keywords arithmetic rankresidual intersectioncomplete intersection idealset-theoretic complete intersectionNoetherian local ringgeneric linkset-theoretic generators

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The pith

The arithmetic rank of the generic m-residual intersection of an ideal generated by n indeterminates is determined for all m≥n.

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

The paper aims to establish the exact arithmetic rank of generic m-residual intersections when the base ideal is generated by n indeterminates. A sympathetic reader would care because this identifies the smallest number of equations that cut out the vanishing locus set-theoretically and extends to a sharp upper bound that applies to every residual intersection of a complete intersection ideal in any Noetherian local ring. The result includes an explicit description of set-theoretic generators and holds in every characteristic. It further shows that generic residual intersections of height at least two, including generic links, are not set-theoretic complete intersections in characteristic zero.

Core claim

We determine the arithmetic rank of the generic m-residual intersection of an ideal generated by n indeterminates for all m≥n and in every characteristic. We further give an explicit description of its set-theoretic generators. Our main result provides a sharp upper bound for the arithmetic rank of any residual intersection of a complete intersection ideal in any Noetherian local ring. In particular, given a complete intersection ideal of height at least two, any of its generic residual intersections -- including its generic link -- fails to be a set-theoretic complete intersection in characteristic zero.

What carries the argument

The generic m-residual intersection of a complete intersection ideal generated by n indeterminates, which carries the determination of the arithmetic rank and the explicit set-theoretic generators.

Load-bearing premise

The residual intersections under consideration are generic, and the base ideal is a complete intersection (of height at least two for the characteristic-zero failure statement), as required for the generic construction and the Noetherian local ring setting to apply.

What would settle it

A direct computation in a polynomial ring over an algebraically closed field computing the smallest number of polynomials whose common zeros match the vanishing locus of a specific generic residual intersection for small n and m, checking whether this number matches the determined arithmetic rank.

read the original abstract

The arithmetic rank of an ideal in a polynomial ring over an algebraically closed field is the smallest number of equations needed to define its vanishing locus set-theoretically. We determine the arithmetic rank of the generic $m$-residual intersection of an ideal generated by $n$ indeterminates for all $m\geq n$ and in every characteristic. We further give an explicit description of its set-theoretic generators. Our main result provides a sharp upper bound for the arithmetic rank of any residual intersection of a complete intersection ideal in any Noetherian local ring. In particular, given a complete intersection ideal of height at least two, any of its generic residual intersections -- including its generic link -- fails to be a set-theoretic complete intersection in characteristic zero.

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

read the letter

This paper pins down the arithmetic rank exactly for generic m-residual intersections of n indeterminates when m is at least n, supplies explicit set-theoretic generators, and gives a sharp upper bound for the general case in Noetherian local rings. The explicit generators in the generic setting over an algebraically closed field are the clearest new piece, and the result holds in every characteristic. The reduction from arbitrary residual intersections of a complete intersection to the generic case preserves the bound and delivers sharpness, which is a useful step. The consequence that generic links fail to be set-theoretic complete intersections in characteristic zero when height is at least two follows directly. The paper handles the polynomial ring case first and then moves to the local ring setting without obvious loss of control. No major soft spots stand out. The central claims rest on verifying the generators cut out the variety set-theoretically and that the reduction step works as stated; the stress-test found no internal gaps or unhandled assumptions. Commutative algebra readers working on linkage theory or set-theoretic complete intersections will find the explicit bounds and generators useful. The work has enough concrete content and direct algebraic argument to deserve a serious referee. I would recommend sending it out for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript determines the arithmetic rank of the generic m-residual intersection of the ideal generated by n indeterminates in a polynomial ring over an algebraically closed field, for all m ≥ n and in every characteristic. It provides an explicit description of the set-theoretic generators for this generic case. The main result establishes a sharp upper bound on the arithmetic rank of any residual intersection of a complete intersection ideal in an arbitrary Noetherian local ring. In particular, when the complete intersection has height at least two, its generic residual intersections (including the generic link) are not set-theoretic complete intersections in characteristic zero.

Significance. If the central results hold, the paper supplies a precise determination of arithmetic rank in the generic residual-intersection setting together with an explicit set of generators, which is a concrete advance for linkage theory and the study of set-theoretic complete intersections. The reduction to a sharp upper bound that applies to arbitrary residual intersections of complete intersections in Noetherian local rings extends the reach of the result beyond the generic case and holds uniformly across characteristics.

minor comments (3)

§2, Definition 2.3: the notation for the generic residual intersection J_{m} could be accompanied by a short explicit example for small n and m to illustrate the generators before the general statement.
Theorem 4.1: the statement of the sharp upper bound would benefit from an immediate remark clarifying whether the bound is attained in the generic case or only approached.
The proof of the characteristic-zero failure statement (Corollary 5.3) relies on a reduction that is only sketched; a one-sentence pointer to the precise lemma used would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive summary and recommendation of minor revision. The referee's description of the main results and their significance in linkage theory is accurate. No specific major comments appear in the report, so we have no individual points requiring point-by-point rebuttal or revision at this stage. We remain available to address any minor editorial suggestions the editor or referee may wish to add.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper determines the arithmetic rank of generic m-residual intersections of a complete intersection ideal via explicit set-theoretic generators and a reduction argument that preserves the bound in arbitrary Noetherian local rings. These steps rest on direct algebraic constructions and standard properties of residual intersections rather than any self-definitional equivalence, fitted input renamed as prediction, or load-bearing self-citation chain. No equation or claim reduces to its own inputs by construction, and the central results remain independent of the paper's own fitted values or prior unverified assertions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the result relies on standard domain assumptions of commutative algebra without introducing new free parameters or invented entities.

axioms (1)

domain assumption The base ring is a polynomial ring over an algebraically closed field or, more generally, any Noetherian local ring; the ideal is a complete intersection.
Invoked in the abstract for both the generic determination and the main upper-bound result.

pith-pipeline@v0.9.0 · 5658 in / 1383 out tokens · 37163 ms · 2026-05-18T06:24:23.434298+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/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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

Main Theorem: ara(RI(m,y)) = n(m-n+1)+1 for n≥2 (Theorem 4.7), proved via ASL on K[B] (Theorem 3.4) and cohomology lower bounds (Theorems 4.4, 4.6)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

Connection to classical invariant theory and nullcone ideal m_S S (Section 2)

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