Generic Initial Ideals and Graded Betti Numbers
Generic Initial Ideals and Graded Betti Numbers
复制标题
通用初始理想和分级贝蒂数
DOI：
10.2969/aspm/03310075
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
J. Herzog
中科院分区：
文献类型：
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作者：
J. Herzog
本文的目的是给出 Kalai [26]、[27] 提出的移位理论的代数背景。对理论的组合方面感兴趣的读者应该查阅 Kalai 的调查论文 [26] 和他在本卷中的文章。在本文中，我们主要感兴趣的是分级贝蒂数在代数移位运算下的行为。代数平移与泛型初始理想理论密切相关。在第一节中，我们回顾了该理论的一些基本事实。下一节致力于研究稳定理想和强稳定理想，因为通用初始理想就是这种类型，前提是基场的特征为 0。在第 3 节中，我们描述了稳定理想和无平方稳定理想的 Betti 数，在第 4 节中，Cartan 复形提供了外代数留数类域的分级最小自由分辨率。对于组合应用具有重要意义的无平方单项式理想理论，有必要研究分级理想、分级模及其在外代数上的解析。在第 5 节中，我们解释了外代数和对称代数上的无平方单项式理想的分级 Betti 数是如何相关的。以下两节专门介绍 Bayer、Charalambous 和 S. Popescu [12] 对极值 Betti 数定理的证明，以及 Aramova 和作者 [4] 在无平方情况下的相应定理。在第 8 节中，我们描述了各种移位运算符并应用前面各节的同调理论。第 9 节应用对称代数移位以及 Bjorner 和 Kalai [15] 定理来推导超极值 Betti 数定理。在最后一节中，简要概述了词段理想的极值属性。
The purpose of this article is to give the algebraic background of the shifting theory developed by Kalai [26], [27]. The reader who is interested in the combinatorial aspects of the theory should consult Kalai's survey paper [26] and his article in this volume. In the present article we are mainly interested in the behaviour of graded Betti numbers under the operation of algebraic shifting. Algebraic shifting is intimately related to the theory of generic initial ideals. In Section 1 we recall some of the basic facts of this theory. The next section is devoted to the study of stable and strongly stable ideals since generic initial ideals are of this kind, provided the base field is of characteristic 0. In Section 3 we describe the Betti numbers of stable and squarefree stable ideals, and in Section 4 the Cartan complex which provides the graded minimal free resolution of the residue class field of the exterior algebra. For the theory of squarefree monomial ideals, which is significant for combinatorial applications, it is necessary to study graded ideals, graded modules and their resolutions over the exterior algebra. In Section 5 we explain how the graded Betti numbers of squarefree monomial ideals over the exterior and symmetric algebra are related. The following two sections are devoted to the proof of a theorem on extremal Betti numbers by Bayer, Charalambous and S. Popescu [12], as well as to the corresponding theorem in the squarefree case by Aramova and the author [4]. In Section 8 we describe various shifting operators and apply the homological theory of the previous sections. Symmetric algebraic shifting and a theorem of Bjorner and Kalai [15] are applied in Section 9 in order to deduce a theorem on superextremal Betti numbers. In the final section extremality properties of lexsegment ideals are briefly sketched.
DOI：
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发表时间：
2006
期刊：
Integrable systems, geometry, and topology, AMS/IP Studies of Advanced Mathematics, American Mathematical Society 36
影响因子：
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作者：
FURUYA;Jun;Martin Guest
通讯作者：
Martin Guest