Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A characterization of binomial Macaulay dual generators for complete intersections

Kohsuke Shibata

TL;DR

The paper characterizes binomial Macaulay dual generators $F$ for which the Artinian algebra $R/\operatorname{Ann}_R(F)$ is a complete intersection, delivering necessary and sufficient conditions in terms of the binomial's support parameters $d_1,d_2$ and an integer $v$ defined via the Macaulay action. It provides explicit descriptions of $\operatorname{Ann}_R(F)$ in the CI cases and shows non-CI behavior when multiple nonzero $b_i$ occur, thereby completing the classification for binomial $F$. As an application, the authors prove that, in characteristic zero, any homogeneous binomial $F$ yielding a complete intersection algebra satisfies the strong Lefschetz property. These results extend earlier monomial and low-dimensional cases, offering a comprehensive understanding of binomial Macaulay dual generators in the complete-intersection regime and their Lefschetz behavior.

Abstract

We characterize a binomial such that the Artinian algebra whose Macaulay dual generator is the binomial is a complete intersection. As an application, we prove that the Artinian algebra with a binomial Macaulay dual generator has the strong Lefschetz property in characteristic 0 if the Artinian algebra is a complete intersection.

A characterization of binomial Macaulay dual generators for complete intersections

TL;DR

The paper characterizes binomial Macaulay dual generators for which the Artinian algebra is a complete intersection, delivering necessary and sufficient conditions in terms of the binomial's support parameters and an integer defined via the Macaulay action. It provides explicit descriptions of in the CI cases and shows non-CI behavior when multiple nonzero occur, thereby completing the classification for binomial . As an application, the authors prove that, in characteristic zero, any homogeneous binomial yielding a complete intersection algebra satisfies the strong Lefschetz property. These results extend earlier monomial and low-dimensional cases, offering a comprehensive understanding of binomial Macaulay dual generators in the complete-intersection regime and their Lefschetz behavior.

Abstract

We characterize a binomial such that the Artinian algebra whose Macaulay dual generator is the binomial is a complete intersection. As an application, we prove that the Artinian algebra with a binomial Macaulay dual generator has the strong Lefschetz property in characteristic 0 if the Artinian algebra is a complete intersection.

Paper Structure

This paper contains 3 sections, 11 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. binomial Macaulay dual generator for c.i
  3. the strong Lefschetz property

Key Result

Theorem 1.3

Let $m,n\in \mathbb N$. Let $R=k[x_1,\dots,x_{m+n}]$ and $S=k[X_1,\dots,X_{m+n}]$ be polynomial rings. Let be a binomial, where $a_1,\dots,a_{m+n},b_1\dots,b_{m+n}\in \mathbb Z_{\ge 0}$ and $c_1,c_2\in k\setminus \{0\}$. Let Suppose that $d_1\ge d_2\ge 1$, Let Then

Theorems & Definitions (28)

  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Claim 2.6
  • ...and 18 more