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

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

TL;DR

The paper determines the arithmetic rank of the generic $m$-residual intersection of a complete intersection, showing $\operatorname{ara}(\mathrm{RI}(m,\underline{y}))=n(m-n+1)+1$ for all $m\ge n$ and all characteristics, and provides an explicit set of generators up to radical. It establishes an upper bound via constructing an auxiliary ring $K[B]$ with an Algebra with a Straightening Law (ASL) structure, computes its dimension $n(m-n+1)+1$, and proves additional structural properties (Gorenstein factorial domain; $F$-regular in positive characteristic). In characteristic zero, a local cohomology obstruction attached to invariant theory forces the equality and implies that generic residual intersections of a complete intersection with height at least two are not set-theoretic complete intersections. The work thereby connects residual intersections to $\mathrm{SL}_n$-invariant theory, uses ASL techniques to obtain explicit generators, and employs topological/étale cohomology to derive sharp lower bounds, yielding a comprehensive, characteristic-free understanding of the arithmetical structure of these ideals.

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.

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

TL;DR

The paper determines the arithmetic rank of the generic -residual intersection of a complete intersection, showing for all and all characteristics, and provides an explicit set of generators up to radical. It establishes an upper bound via constructing an auxiliary ring with an Algebra with a Straightening Law (ASL) structure, computes its dimension , and proves additional structural properties (Gorenstein factorial domain; -regular in positive characteristic). In characteristic zero, a local cohomology obstruction attached to invariant theory forces the equality and implies that generic residual intersections of a complete intersection with height at least two are not set-theoretic complete intersections. The work thereby connects residual intersections to -invariant theory, uses ASL techniques to obtain explicit generators, and employs topological/étale cohomology to derive sharp lower bounds, yielding a comprehensive, characteristic-free understanding of the arithmetical structure of these ideals.

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 -residual intersection of an ideal generated by indeterminates for all 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.
Paper Structure (8 sections, 21 theorems, 99 equations)

This paper contains 8 sections, 21 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. A curious connection with classical invariant theory
  3. Upper bound on the arithmetic rank: Structure of the auxiliary ring
  4. The auxiliary ring is an ASL
  5. The auxiliary ring is a Gorenstein factorial domain
  6. The auxiliary ring is F-regular
  7. Lower bound on the arithmetic rank: Topology of the residual intersection
  8. A second take on the dimension of the auxiliary ring

Key Result

Proposition 2.1

Let $K$ be a field of characteristic zero. Then

Theorems & Definitions (46)

  • Definition 1.1
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Example 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • proof
  • ...and 36 more