Filter-regular sequences, almost complete intersections and Stanley's conjecture

TL;DR

This paper investigates conditions under which monomial ideals lead to pretty clean quotient rings, proving Stanley's conjecture and related properties for these classes, including almost complete intersections and certain simplicial complexes.

Contribution

It establishes that monomial ideals generated by filter-regular sequences, d-sequences, or almost complete intersections produce pretty clean, sequentially Cohen-Macaulay quotients, confirming Stanley's conjecture in these cases.

Findings

$S/I$ is pretty clean under specified conditions.

Stanley's and $h$-regularity conjectures hold for these classes.

Stanley's conjecture holds for Stanley-Reisner ideals of locally complete intersection complexes.

Abstract

Let $K$ be a field and $I$ a monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$ generated by monomials $u_1,u_2,..., u_t$. We show that $S/I$ is pretty clean if either: 1) $u_1,u_2,..., u_t$ is a filter-regular sequence, 2) $u_1,u_2,..., u_t$ is a $d$-sequence; or 3) $I$ is almost complete intersection. In particular, in each of these cases, $S/I$ is sequentially Cohen-Macaulay and both Stanley's and $h$-regularity conjectures, on Stanley decompositions, hold for $S/I$. Also, we prove that if $I$ is the Stanley-Reisner ideal of a locally complete intersection simplicial complex on $[n]$, then Stanley's conjecture holds for $S/I$.