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On the Stanley depth of powers of some classes of monomial ideals

Mircea Cimpoeas

Abstract

Given arbitrary monomial ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $K$, we investigate the Stanley depth of powers of the sum $I+J$, and their quotient rings, in $A\otimes_K B$ in terms of those of $I$ and $J$. Our results can be used to study the asymptotic behavior of the Stanley depth of powers of a monomial ideal. For instance, we solved the case of monomial complete intersection.

On the Stanley depth of powers of some classes of monomial ideals

Abstract

Given arbitrary monomial ideals and in polynomial rings and over a field , we investigate the Stanley depth of powers of the sum , and their quotient rings, in in terms of those of and . Our results can be used to study the asymptotic behavior of the Stanley depth of powers of a monomial ideal. For instance, we solved the case of monomial complete intersection.

Paper Structure

This paper contains 2 sections, 16 theorems, 16 equations.

Table of Contents

  1. Preliminaries
  2. Main results

Key Result

Lemma 1.3

(Depth Lemma) If $0 \rightarrow U \rightarrow M \rightarrow N \rightarrow 0$ is a short exact sequence of modules over a local ring $S$, or a Noetherian graded ring with $S_0$ local, then a) $\operatorname{depth} M \geq \min\{\operatorname{depth} N,\operatorname{depth} U\}$. b) $\operatorname{depth}

Theorems & Definitions (26)

  • Conjecture 1.1
  • Conjecture 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • ...and 16 more