On minimal graded free resolutions

On minimal graded free resolutions
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关于最小分级自由分辨率

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发表时间：
2001
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通讯作者：
Tim Römer
Tim Römer
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作者：
Tim Römer

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最小分次自由分解是代数中一个重要的中心问题。 它们是研究有限生成分次K-上的模的有用工具。 代数。这样的分辨率决定了希尔伯特系列，卡斯特尔诺沃-芒福德 模的正则性和其他不变量。 本文主要研究最小分次自由分解的结构。我们 将我们的结果与交换代数中的几个最新趋势联系起来。 这些趋势中的第一个涉及赤柱属性之间的关系- 单纯复形的Reisner环及其Stanley-Reisner环 亚力山大双打。 另一个发展是对极小分次图的线性部分的研究 Eisenbud和Schreyer定义的自由分辨率。 几位作者对这个问题感兴趣，以便给出Betti的下界 模块的编号。特别是，Eisenbud-Koh，Green，Herzog和Reiner- 韦尔克研究了分次Betti数，它决定了一条线性链 最小分级自由分辨率。 大格代数自然地出现在交换代数的许多研究领域中。 二重代数的一个典型例子是分次理想的Rees环。赫索格 Trung利用Rees环的这种粗大结构来研究Castelnuovo- 多项式环中分次理想的幂的Mumford正则性。孔卡，赫尔佐格， Trung和Valla研究了双格化代数的对角子代数。阿拉莫娃 Crona和De Negri研究了二重K-代数的同调性质。
Minimal graded free resolutions are an important and central topic in algebra. They are a useful tool for studying modules over finitely generated graded K- algebras. Such a resolution determines the Hilbert series, the Castelnuovo-Mumford regularity and other invariants of the module. This thesis is concerned with the structure of minimal graded free resolutions. We relate our results to several recent trends in commutative algebra. The first of these trends deals with relations between properties of the Stanley- Reisner ring associated to a simplicial complex and the Stanley-Reisner ring of its Alexander dual. Another development is the investigation of the linear part of a minimal graded free resolution as defined by Eisenbud and Schreyer. Several authors were interested in the problem to give lower bounds for the Betti numbers of a module. In particular, Eisenbud-Koh, Green, Herzog and Reiner- Welker studied the graded Betti numbers which determine the linear strand of a minimal graded free resolution. Bigraded algebras occur naturally in many research areas of commutative algebra. A typical example of a bigraded algebra is the Rees ring of a graded ideal. Herzog and Trung used this bigraded structure of the Rees ring to study the Castelnuovo- Mumford regularity of powers of graded ideals in a polynomial ring. Conca, Herzog, Trung and Valla dealt with diagonal subalgebras of bigraded algebras. Aramova, Crona and De Negri studied homological properties of bigraded K-algebras.