Stanley-reisner rings with pure resolutions
Stanley-reisner rings with pure resolutions
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Stanley-reisner 戒指具有纯粹的分辨率
DOI：
10.1080/00927879508825274
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发表时间：
1995
影响因子：
0.7
通讯作者：
T. Hibi
中科院分区：
文献类型：
--
作者：
W. Bruns;T. Hibi
在这篇文章中，我们继续我们在[3]中开始的关于单复形A的研究，它的Stanley-Reisner环有纯极小自由分解。(多项式环上的自由模的复形称为纯的，如果它的映射可以用矩阵p表示，其所有非零项都具有相同的次数，这可能取决于i)。虽然努力实现它们的完全组合分类可能过于雄心勃勃，但寻找由简单复合体产生的所有纯自由分辨率的数值不变量似乎是一个合理的计划，至少在Cohen-Macaulay性质存在的情况下是这样。一个非常令人满意的情况是A是由它的1骨架A1支撑的最大复数，或者等价地，其中Stanley-Reisner环K[A]的定义理想I*是由2次单项式生成的(K是任意域)。Froberg[5]证明了K[A]有2-线性分解当且仅当Al是弦图，我们证明了在所有其他情况下，在纯分解的情况下，A是i维圈上的多锥图。(如果对所有WCV有#W=3，使得RW是圈，则顶点集V上的图r是弦图。)特别地，对具有纯自由分辨率的一维单纯复形以及由偏序集产生的复形进行了完全分类。接下来我们讨论2维的情形。主要的困难情形是ID由3次单项式产生，并且A是双重Cohen-Macaulay的。对于每一数目n的顶点，至多存在一种数值类型的分辨率，
In this paper we continue our study, started in [3], of simplicial complexes A whose Stanley-Reisner rings have pure minimal free resolutions.(A complex of free modules over a polynomial ring is calledpure if its maps can be represented by matrices p, all of whose non-zero entries have the same degree, which may depend on i). While it might be too ambitious to strive for their complete combinatorial classification, it seems to be a reasonable project to find the numerical invariants of all pure free resolutions that arise from simplicia1 complexes, at least in the presence of the Cohen-Macaulay property. A very satisfactory case is that in which A is the maximal complex supported by its 1-skeleton A 1, or, equivalently, in which the defining ideal I* of the Stanley-Reisner ring K [A] is generated by monomials of degree 2 (K is an arbitrary field). Froberg [5] has shown that K [A] has a 2-linear resolution if and only if Al is a chordal graph, and we complement his result by proving that in all the other cases with a pure resolution A is a multi-cone over a I-dimensional cycle.(A graph r on the vertex set V is chordal if# W= 3 for all Wc V such that rw is a cycle.) In particular the 1-dimensional simplicial complexes with pure free resolutions are completely classified, as well as those arising from a partially ordered set. We next address the case of dimension 2. The main difficult case is that in which Id is generated by degree 3 monomials and A is doubly Cohen-Macaulay. For each number n of vertices there exists at most one numerical type of resolution,