论文标题
简单子隔板的多层最小免费分辨率
Multigraded minimal free resolutions of simplicial subclutters
论文作者
论文摘要
本文涉及对称为简单子隔板的一类剪裁器的研究。给定一个混乱$ \ Mathcal {C} $及其简单的subclutter $ \ Mathcal {d} $，我们分别比较了与这两个杂物相关的理想$ i，j $的一些代数属性和不变性。我们提供了一个公式，用于计算$ i $的（$ i $和一些组合数据的$ j $）的（多）分级的betti数字。结果，我们看到，如果$ \ Mathcal {c} $承认一个简单的子集机，则存在一个单一的$ u \ notin i $，因此可以通过$ i $的$ $ $ $ $ $计算（多）分级的betti betti $ i+（u）$。事实证明，任何具有线性分辨率的分级理想的贝蒂序列是与完整杂物的简单子集合相关的理想序列的贝蒂序列。这些理想证明具有线性商。但是，它们并未与线性商形成所有均衡的无正方形理想。如果$ \ MATHCAL {C} $作为简单的$ \ varnothing $作为简单的子关键，则$ i $在所有字段上都具有线性分辨率。示例表明，相反是不正确的。
This paper concerns the study of a class of clutters called simplicial subclutters. Given a clutter $\mathcal{C}$ and its simplicial subclutter $\mathcal{D}$, we compare some algebraic properties and invariants of the ideals $I, J$ 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 $\mathcal{D}$. As a result, we see that if $\mathcal{C}$ admits a simplicial subclutter, then there exists a monomial $u \notin I$ such that the (multi)graded Betti numbers of $I+(u)$ 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 $\mathcal{C}$ admits $\varnothing$ as a simplicial subclutter, then $I$ has linear resolution over all fields. Examples show that the converse is not true.
