ON THE EXT-MODULES OF IDEALS OF BOREL TYPE
ON THE EXT-MODULES OF IDEALS OF BOREL TYPE
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论Borel型理想的外模
DOI：
10.1090/conm/331/05909
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Marius Vladoiu
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文献类型：
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作者：
Marius Vladoiu
Bayer和Stillman（参见[3，命题15.24]）证明了Borel-不动理想是Borel型的。我们的第一个结果，证明在第1节，说理想的博雷尔型是顺序科恩-麦考利。这推广了[5]中的观察，即由强稳定理想定义的环是序列Cohen-Macaulay环。强稳定理想是Borel-固定的，因此是Borel型理想。序列Cohen-Macaulay模的概念最早是由Stanley [8]在组合背景下引入的。Schenzel [7]独立地定义了这样的模，称它们为Cohen-Macaulay滤模，并用同调的术语刻画了它们。设R = S/I是序列Cohen-Macaulay的标准分次K-代数.然后，连接到顺序Cohen-Macaulay环R的自然过滤允许根据该过滤明确地计算Ext-modules ExtS（R，S），参见。推论2.6。特别是，这种技术允许在情况下，我是博雷尔型，以确定ai-不变量的R作为基座度的某些有限长度的模块，如推论2.7所示。这些方法在下面的章节中被应用，见推论3.3，以确定主p-Borel理想的ai-不变量。这一结果改进了Pardue提出并由Aramova，Herzog和Popescu在[1]和[4]中证明的主p-Borel理想的正则性公式.它还提供了一个更优雅和透明的证明帕杜公式。这种新方法的另一个优点是定理4.4和推论4.6中的柱脚计算通过推论2.6产生了模ExtS（R，S）的生成元的最小集合。当I是S的m-准素理想时，则在推论4.3中计算R的柱脚维数。作者感谢亚历山大冯洪堡基金会支持他们的合作。
Bayer and Stillman (cf. [3, Proposition 15.24]) proved that Borel-fixed ideals are of Borel type. Our first result, proved in Section 1, says that ideals of Borel type are sequentially Cohen-Macaulay. This generalizes the observation made in [5], that rings defined by strongly stable ideals are sequentially Cohen-Macaulay. Strongly stable ideals are Borel-fixed and hence are ideals of Borel type. The concept of sequentially Cohen-Macaulay modules was first introduced in combinatorial contexts by Stanley [8]. Independently Schenzel [7] defined such modules, called them Cohen-Macaulay filtered module, and characterized them in homological terms. Let R = S/I a standard graded K-algebra which is sequentially Cohen-Macaulay. Then the natural filtration attached to the sequentially Cohen-Macaulay ring R allows the compute the Ext-modules ExtS(R, S) explicitly in terms of this filtration, cf. Corollary 2.6. In particular, this technique permits in case I is of Borel type to determine the ai-invariants of R as socle degrees of certain finite length modules, as is shown in Corollary 2.7. These methods are applied in the following sections, see Corollary 3.3, to determine the ai-invariants of principal p-Borel ideals. This result refines the formula for the regularity of principal p-Borel ideals which was conjectured by Pardue and proved by Aramova, Herzog and Popescu in [1] and [4]. It also provides a more elegant and transparent proof of the Pardue formula. Another advantage of this new approach is that the socle computation in Theorem 4.4 and Corollary 4.6 yields, via Corollary 2.6, a minimal set of generators of the modules ExtS(R, S). When I is an m-primary ideal of S, then in Corollary 4.3 the socle dimension of R is computed. The authors are grateful to the Alexander von Humboldt Foundation for supporting their collaboration.