Weakly Polymatroidal Ideals

Abstract

The concept of the weakly polymatroidal ideal, which generalizes both the polymatroidal ideal and the prestable ideal, is introduced. A fundamental fact is that every weakly polymatroidal ideal has a linear resolution. One of the typical examples of weakly polymatroidal ideals arises from finite partially ordered sets. We associate each weakly polymatroidal ideal with a finite sequence, called the polymatroidal sequence, which will be useful for the computation of graded Betti numbers of weakly polymatroidal ideals as well as for the construction of weakly polymatroidal ideals.

Monomial ideals and toric rings of Hibi type arising from a finite poset

“…Based on this explicit description we show in Theorem 2.2 that all powers H r (P ) k of the ideal H r (P ) are weakly polymatroidal. The concept of weakly polymatroidal ideals has been introduced by Hibi and Kokubo in [15] where they showed that they share with polymatroidal ideals the nice property of having linear quotients. In particular, we conclude from this that the ideals H r (P ) k have a linear resolution for all k ≥ 1.…”

Powers of the vertex cover ideals

“…, x n } for all i. Ideals with linear quotients were introduced by Herzog and Takayama in [12]. A class of ideals enjoying the nice property of having linear quotients is the class of weakly polymatroidal ideals introduced by Hibi and Kokubo in [13] for ideals generated in the same degree (and later in [15] for ideals not necessarily generated in one degree).…”

Recursive Betti numbers for Cohen–Macaulay d -partite clutters arising from posets

“…In [7,Theorem 2.4], the authors proved that H P,d is a weakly polymatroidal ideal. Weakly polymatroidal ideals have been introduced by Hibi and Kokubo in [14]. From [26,Theorem 1.3], it follows that a weakly polymatroidal ideal has linear quotients.…”