[Paper Review] Persistence and stability properties of powers of ideals

Jürgen Herzog, Ayesha Asloob Qureshı|arXiv (Cornell University)|Aug 23, 2012

Commutative Algebra and Its Applications15 references4 citations

TL;DR

This paper introduces the concept of strong persistence for ideals, showing it implies persistence of associated prime ideals in powers of an ideal and is equivalent to Ratliff's condition $I^{k+1}:I = I^k$ for all $k$. The key contribution is proving that polymatroidal ideals satisfy strong persistence, leading to bounds on the index of depth stability and associated prime stability in terms of analytic spread.

ABSTRACT

We introduce the concept of strong persistence and show that it implies persistence regarding the associated prime ideals of the powers of an ideal. We also show that strong persistence is equivalent to a condition on power of ideals studied by Ratliff. Furthermore, we give an upper bound for the depth of powers of monomial ideals in terms of their linear relation graph, and apply this to show that the index of depth stability and the index of stability for the associated prime ideals of polymatroidal ideals is bounded by their analytic spread.

Motivation & Objective

• To define and analyze the strong persistence property for ideals, which strengthens the persistence property for associated prime ideals.
• To establish that strong persistence is equivalent to Ratliff's condition $I^{k+1}:I = I^k$ for all $k$.
• To prove that polymatroidal ideals satisfy strong persistence, thereby bounding their index of depth stability and associated prime stability.
• To provide an upper bound for the depth of powers of monomial ideals using their linear relation graph.
• To show that for polymatroidal ideals, both the index of depth stability and the index of associated prime stability are bounded by the analytic spread.

Proposed method

• Introduce the concept of strong persistence via localization and socle element behavior under multiplication by generators of $I$.
• Prove that strong persistence is equivalent to the condition $I^{k+1}:I = I^k$ for all $k \geq 1$, linking it to Ratliff's work.
• Use the Rees algebra $\mathcal{R}(I)$ satisfying Serre's condition $S_2$ to deduce strong persistence for ideals of positive grade.
• Analyze the linear relation graph $\Gamma$ of monomial ideals to bound the depth of $S/I^k$.
• Characterize the connected components of $\Gamma$ for transversal polymatroidal ideals as complete graphs on vertex sets of connected components of the simplicial complex $\Delta$.
• Apply these results to Veronese-type polymatroidal ideals, showing $\Gamma$ is connected and $\ell(I) = \operatorname{supp}(I)$.

Experimental results

Research questions

• RQ1 Does the strong persistence property imply the persistence property for associated prime ideals of powers of an ideal?
• RQ2 Is the condition $I^{k+1}:I = I^k$ for all $k$ equivalent to strong persistence?
• RQ3 Can the index of depth stability and the index of associated prime stability be bounded for polymatroidal ideals?
• RQ4 How does the linear relation graph of a monomial ideal relate to the depth of its powers?
• RQ5 What is the analytic spread of a transversal polymatroidal ideal, and how does it relate to stability indices?

Key findings

• Strong persistence implies persistence of associated prime ideals and is equivalent to the condition $I^{k+1}:I = I^k$ for all $k$.
• Polymatroidal ideals satisfy strong persistence, hence their associated prime ideals stabilize and their depth functions are non-increasing.
• The index of depth stability $\operatorname{dstab}(I)$ and the index of associated prime stability $\operatorname{astab}(I)$ are both bounded above by the analytic spread $\ell(I)$ for polymatroidal ideals.
• For transversal polymatroidal ideals, the linear relation graph $\Gamma$ has connected components corresponding to the connected components of the simplicial complex $\Delta$, each being a complete graph.
• For Veronese-type polymatroidal ideals, the linear relation graph $\Gamma$ is connected and $\ell(I) = \operatorname{supp}(I)$, the set of variables dividing some generator.
• The depth function $\operatorname{depth}S/I^k$ is non-increasing for all polymatroidal ideals, and stabilizes for $k \gg 0$.