Classes of sequentially Cohen-Macaulay squarefree monomial ideals (Q2928441)

Let \(S=k[x_1,\dots, x_n]\) (\(k\) field) be a polynomial ring where the monomials are ordered lexicographically with \(x_1> \dots > x_n\). If \(u,v \in S\) are squarefree monomials of the same degree \(q\), the set \(L(u,v)=\{w \in S \mid w \text{ squarefree monomial of degree } q \text{ and } u \geq w \geq v \}\) is called a squarefree lexsegment of degree \(q\). For an arbitrary set of monomials \(T\) in \(S\) we denote \(\text{Shad} T = \{ wx_i \mid w \in T, i=1,\dots, n \text{ and } x_i \text{ does not divide } w \}\) and recursively \(\text{Shad}^i T=\text{Shad}^{i-1}(\text{Shad}T)\). A squarefree lexsegment \(L=L(u,v)\) is called completely squarefree lexsegment if \(\text{Shad}^i L\) is a squarefree lexsegment for all \(i \geq 1\).NEWLINENEWLINEIn this paper the author (1) obtains the minimal primary decomposition for a completely squarefree lexsegment \(L\); (2) describes \(\text{dim} S /I\) when \(I\) is a completely squarefree lexsegment ideal; (3) obtains the multiplicity of \(k[\Delta]\) where \(\Delta\) is the simplicial complex with Stanley-Reisner ideal \(I\); (4) gives a description of the completely squarefree lexsegment ideals that are sequentially Cohen-Macaulay; and (5) gives upper and lower bounds for \(\text{depth} S/I\) for an arbitrary squarefree lexsegment ideal \(I\).