Sequentially Cohen-Macaulay and pretty clean monomial ideals

TL;DR

The paper addresses when $R/I$ is pretty clean versus sequentially Cohen-Macaulay for monomial ideals, focusing on generic monomial ideals. Using prime filtrations, the Herzog-Popescu framework, and Alexander duality, the authors relate sequentially Cohen-Macaulayness to the dual being componentwise linear, proving that for generic $I$, $R/I$ is pretty clean iff $R/I$ is sequentially Cohen-Macaulay. They extend the equivalence to certain special classes of monomial ideals and provide a counterexample showing that a conjecture by Soleyman Jahan that all generic monomial ideals are pretty clean is false. Significance: links combinatorial data of generator sets to homological properties of monomial ideals, clarifying when sequentially Cohen-Macaulayness and pretty cleanliness coincide and highlighting limits of existing conjectures.

Abstract

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be monomial ideal of $R$. In this paper, we show that if $I$ is a generic monomial ideal, then $R/I$ is pretty clean if and only if $R/I$ is sequentially Cohen-Macaulay. Furthermore, we prove that this equivalence remains unchanged for some special monomial ideals. Moreover, we provide an example that disproves the conjecture raised in \cite[p. 123]{S1} regarding generic monomial ideals.

Theorems & Definitions (26)

• Definition 1.1
• Theorem 1.2
• Definition 2.1
• Example 2.2
• Lemma 2.3
• proof
• Corollary 2.4
• Corollary 2.5
• proof
• Theorem 2.6