Algebraic Properties of Universal Squarefree Lexsegment Ideals

Let K be a field and let A=K[X1,…,Xn] be the polynomial ring in X1,…,Xn with coefficients in K. In this paper we study the universal squarefree lexsegment ideals, and put our attention on their combinatorics computing some invariants. Moreover, we study the link between such a special class of squarefree lexsegment ideals and the so-called s-sequences.

Normally Torsion-free Lexsegment Ideals

In this paper we characterize all the lexsegment ideals which are normally torsion-free. This will provide a large class of normally torsion-free monomial ideals which are not square-free. Our characterization is given in terms of the ends of the lexsegment. We also prove that, for lexsegment ideals, the property being normally torsion-free is equivalent to the property of the depth function being constant.

Classes of Sequentially Cohen-Macaulay Squarefree Monomial Ideals

We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen-Macaulay. As an application, we give a complete characterization of the completely squarefree lexsegment ideals which are sequentially Cohen-Macaulay and we also derive formulas for some homological invariants of this class of ideals.

Lexsegment Ideals Are Sequentially Cohen-Macaulay

The associated primes of an arbitrary lexsegment ideal I ⊆ S=K[x1,…,xn] 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.

Monomial ideals of minimal depth

Let S be a polynomial algebra over a field. We study classes of monomial ideals (as for example lexsegment ideals) of S having minimal depth. In particular, Stanley's conjecture holds for these ideals. Also we show that if I is a monomial ideal with Ass(S/I) = {P1, P2, ..., Ps} and Pi ⊄ ∑s1=j≠i Pj for all i ∊ [s], then Stanley’s conjecture holds for S/I.