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Complete intersections in binomial and lattice ideals

Hiram H. Lopez, Rafael H. Villarreal

TL;DR

For the family of graded lattice ideals of dimension 1, it is shown that all ideals of this family are binomial set-theoretic complete intersections.

Abstract

For the family of graded lattice ideals of dimension 1, we establish a complete intersection criterion in algebraic and geometric terms. In positive characteristic, it is shown that all ideals of this family are binomial set theoretic complete intersections. In characteristic zero, we show that an arbitrary lattice ideal which is a binomial set theoretic complete intersection is a complete intersection.

Complete intersections in binomial and lattice ideals

TL;DR

For the family of graded lattice ideals of dimension 1, it is shown that all ideals of this family are binomial set-theoretic complete intersections.

Abstract

For the family of graded lattice ideals of dimension 1, we establish a complete intersection criterion in algebraic and geometric terms. In positive characteristic, it is shown that all ideals of this family are binomial set theoretic complete intersections. In characteristic zero, we show that an arbitrary lattice ideal which is a binomial set theoretic complete intersection is a complete intersection.

Paper Structure

This paper contains 3 sections, 16 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Complete intersections
  3. Vanishing ideals over finite fields

Key Result

Theorem 2.2

EisStu If $L$ is a binomial ideal of $S$, then $L$ is a lattice ideal if and only if $t_i$ is a non-zero divisor of $S/L$ for all $i$.

Theorems & Definitions (37)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Definition 2.7
  • ...and 27 more