Journal of Combinatorial Theory Series A ( IF 0.9 ) Pub Date : 2020-10-07 , DOI: 10.1016/j.jcta.2020.105339 Mina Bigdeli , Ali Akbar Yazdan Pour
This paper concerns the study of a class of clutters called simplicial subclutters. Given a clutter and its simplicial subclutter , we compare some algebraic properties and invariants of the ideals associated to these two clutters, respectively. We give a formula for computing the (multi)graded Betti numbers of J in terms of those of I and some combinatorial data about . As a result, we see that if admits a simplicial subclutter, then there exists a monomial such that the (multi)graded Betti numbers of can be computed through those of I. It is proved that the Betti sequence of any graded ideal with linear resolution is the Betti sequence of an ideal associated to a simplicial subclutter of the complete clutter. These ideals turn out to have linear quotients. However, they do not form all the equigenerated square-free monomial ideals with linear quotients. If admits ∅ as a simplicial subclutter, then I has linear resolution over all fields. Examples show that the converse is not true.
中文翻译:
简单子杂波的多级最小自由分辨率
本文涉及一类称为简单次杂波的杂波的研究。考虑到混乱 及其简单的子杂波 ,我们比较理想的一些代数性质和不变量 分别与这两个杂波相关。我们给出一个公式用于计算(多)的分级贝蒂数Ĵ在那些方面我和约一些组合数据。结果,我们看到 承认一个简单的杂波,然后存在一个单项式 这样的(多)分级贝蒂数 可以通过I来计算。证明了具有线性分辨率的任何渐变理想的Betti序列都是与完整杂波的简单子杂波相关联的理想的Betti序列。这些理想结果具有线性商。但是,它们并不能形成所有带有线性商的均等生成的无平方单项理想。如果承认∅是单纯形子杂波,那么我在所有字段上都有线性分辨率。例子表明,事实并非如此。