Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Abstract

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertexweighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen-Macaulay vertexweighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen-Macaulay and satisfy Alexander duality.

Initially regular sequences and depths of ideals

“…It was shown in [23] that the vanishing ideals of these projective spaces can be generated by a set of binomials that form a Gröbner basis whose resulting initial ideal is precisely the edge ideal of a weighted oriented graph. The Cohen-Macaulay property of these ideals was studied in [10,13]. Using our techniques we can construct initially regular sequences that bound the depths of the edge ideals of general weighted oriented graphs.…”

Projective dimension and regularity of edge ideals of some weighted oriented graphs

“…Since the underlying graph G of D is simple, it has not isolated vertex. Thus t 1 = t 2 and I(D) = (x 21 x w 11 11 , x 22 x w 12 12 , . .…”

“…The edge ideal of a vertex-weighted digraph was first introduced by Gimenez et al [11]. Let D = (V, E, w) be a vertex-weighted digraph with the vertex set V = {x 1 , .…”

Normally torsion-free edge ideals of weighted oriented graphs

“…. , t s ] over a ground field K. The edge ideal of D, introduced in [5,18], is the ideal of S given by I(D) := (t i t w j j | (t i , t j ) ∈ E(D)).…”