A Criterion for a Monomial Ideal to have a Linear Resolution in Characteristic 2

Abstract

In this paper we give a necessary and sufficient combinatorial condition for a monomial ideal to have a linear resolution over fields of characteristic 2. We also give a new proof of Fröberg's theorem over fields of characteristic 2.

Chordality of clutters with vertex decomposable dual and ascent of clutters

“…Another difference arises from the concept of Alexander dual which is illustrated in the following example. ∈ Γ, we see that [5] \ 14 = 235 ∈ Γ ∨ . In fact, Note that in the case that ∆ = ∆(C), then since all possible faces of dimension less than d are in ∆, it follows that H i (∆ W ; K) = 0 for all W ⊆ V and i < d − 1.…”

“…The previous simple result shows why the concept of ascent of a clutter can be useful. For example, we show that a main theorem of [5] can simply follow 4.3 and the results of [6]. First we state the needed results of [6] as a lemma.…”

“…In [5,6], the concept of chorded simplicial complexes is defined and it is proved that I ∆ has a (d + 1)-linear resolution over a field of characteristic 2, if and only if ∆ is chorded and it is the clique complex of a d-clutter. We briefly recall this concept.…”

“…We consider ∅, CF-chordal and we say that a non empty d-clutter C is CF-chordal, when C is d-chorded and C + is CF-chordal. It is easy to check that C is CF-chordal if and only if ∆(C) [m] is m-chorded for all m ≥ d. In [5] simplicial complexes with this property are called chorded. Thus the next result is indeed Theorem 18 of [5], which is one of the two main theorems of that paper.…”

“…It is easy to check that C is CF-chordal if and only if ∆(C) [m] is m-chorded for all m ≥ d. In [5] simplicial complexes with this property are called chorded. Thus the next result is indeed Theorem 18 of [5], which is one of the two main theorems of that paper. Before stating this result, it should be mentioned that an example of a clutter C with C dchorded but C + not (d + 1)-chorded, is presented in [5,Example 16].…”