[Paper Review] A note on Stanley conjecture for monomial ideals

Mircea Cimpoeaş|arXiv (Cornell University)|Nov 7, 2011

Commutative Algebra and Its Applications12 references3 citations

TL;DR

This paper proves that Stanley's conjecture holds for monomial ideals and their quotients when the ideal has a small number of generators relative to the depth of the quotient ring and the size of associated prime supports. In particular, it establishes the conjecture for monomial almost complete intersection ideals in polynomial rings over a field.

ABSTRACT

In this paper, we prove that if $I\subset S:=K[x_1,...,x_n]$ is a monomial ideal then $I$ and $S/I$ satisfy the Stanley conjecture when $I$ has a small number of generators, with respect to $\depth(S/I)$ and $\max\{|P|:\;P\in\Ass(S/I)\}$. In particular, if $I$ be a monomial almost complete intersection ideal in $S$, then Stanley's Conjecture holds for $S/I$ and $I$.

Motivation & Objective

• To investigate the validity of Stanley's conjecture for monomial ideals with a limited number of generators.
• To examine the relationship between the number of generators of a monomial ideal and invariants like depth and associated prime supports.
• To establish the conjecture for monomial almost complete intersection ideals in polynomial rings.

Proposed method

• Analyzing the structure of monomial ideals in polynomial rings $ S = K[x_1, \dots, x_n] $ over a field $ K $.
• Using the depth of $ S/I $ and the maximal size of associated prime ideals $ \max\{|P| : P \in \Ass(S/I)\} $ as key invariants.
• Applying combinatorial and algebraic techniques to bound the Stanley depth in terms of these invariants.
• Focusing on ideals with few generators to derive sufficient conditions for the Stanley conjecture to hold.
• Proving the conjecture for almost complete intersection monomial ideals as a special case.

Experimental results

Research questions

• RQ1 Under what conditions on the number of generators does Stanley's conjecture hold for monomial ideals and their quotients?
• RQ2 How do the depth of $ S/I $ and the size of associated prime ideals influence the validity of the Stanley conjecture?
• RQ3 Does the Stanley conjecture hold for monomial almost complete intersection ideals in polynomial rings?

Key findings

• Stanley's conjecture holds for any monomial ideal $ I \subset S $ when the number of generators is small relative to $ \depth(S/I) $ and $ \max\{|P| : P \in \Ass(S/I)\} $.
• The conjecture is confirmed for monomial almost complete intersection ideals in $ S = K[x_1, \dots, x_n] $.
• The result provides a new class of monomial ideals for which the Stanley conjecture is verified, extending known cases.
• The proof relies on structural constraints derived from the number of generators and associated prime information, offering a general criterion.