Sequentially Cohen-Macaulay and pretty clean monomial ideals

TL;DR

This paper explores the relationship between pretty clean and sequentially Cohen-Macaulay properties of monomial ideals, establishing equivalences for generic cases and providing counterexamples to existing conjectures.

Contribution

It proves that for generic monomial ideals, the quotient ring is pretty clean if and only if it is sequentially Cohen-Macaulay, and extends this to certain special ideals.

Findings

Equivalence between pretty clean and sequentially Cohen-Macaulay for generic monomial ideals.

Counterexample to a conjecture on generic monomial ideals.

Extension of equivalence to some special monomial ideals.

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.