Lexsegment ideals are sequentially Cohen-Macaulay

TL;DR

This paper proves that lexsegment ideals lead to modules that are sequentially Cohen-Macaulay, confirming Stanley's conjecture for these cases, by analyzing their associated primes and module properties.

Contribution

It establishes that lexsegment ideals produce modules that are sequentially Cohen-Macaulay and pretty clean, advancing understanding of their algebraic structure.

Findings

Associated primes of lexsegment ideals are characterized.

Modules over lexsegment ideals are shown to be pretty clean.

S/I satisfies Stanley's conjecture.

Abstract

The associated primes of an arbitrary lexsegment ideal $I\subset S=K[x_1,...,x_n]$ are determined. As application it is shown that $S/I$ is a pretty clean module, therefore, $S/I$ is sequentially Cohen-Macaulay and satisfies Stanley's conjecture.