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Stanley depth of quotient of monomial complete intersection ideals

Mircea Cimpoeas

Abstract

We compute the Stanley depth for a particular, but important case, of the quotient of complete intersection monomial ideals. Also, in the general case, we give sharp bounds for the Stanley depth of a quotient of complete intersection monomial ideals. In particular, we prove the Stanley conjecture for quotients of complete intersection monomial ideals.

Stanley depth of quotient of monomial complete intersection ideals

Abstract

We compute the Stanley depth for a particular, but important case, of the quotient of complete intersection monomial ideals. Also, in the general case, we give sharp bounds for the Stanley depth of a quotient of complete intersection monomial ideals. In particular, we prove the Stanley conjecture for quotients of complete intersection monomial ideals.

Paper Structure

This paper contains 2 sections, 15 theorems, 7 equations.

Table of Contents

  1. The case of irreducible ideals
  2. The case of complete intersection ideals

Key Result

Lemma 1.1

Let $b$ be a positive integer, denote $S'=K[x_2,\ldots,x_n]$ and let $I\subsetneq J\subset S'$ be two monomial ideals. Then $(x_1^b,J)/(x_1^b,I) \cong \bigoplus_{i=0}^{b-1} x_1^i(J/I)$, as $\mathbb Z^{n}$-graded $S'$-modules. Moreover, $\operatorname{sdepth}_S((x_1^b,J)/(x_1^b,I)) = \operatorname{sd

Theorems & Definitions (30)

  • Lemma 1.1
  • proof
  • Corollary 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Theorem 1.5
  • proof
  • ...and 20 more