Koszul homology of Cohen-Macaulay rings with linear resolutions
Koszul homology of Cohen-Macaulay rings with linear resolutions
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具有线性分辨率的 Cohen-Macaulay 环的 Koszul 同调
DOI：
10.1090/s0002-9939-1992-1089412-9
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发表时间：
1992
期刊：
影响因子：
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通讯作者：
R. Villarreal
中科院分区：
文献类型：
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作者：
C. Rentería;R. Villarreal
抽象的。研究了具有线性分解的Cohen-Macaulay理想的第一Koszul同调模，并给出了极小多重环的一些新例子. 1.设R是域k上的多项式环，f是R的分次理想。代数S = R/I是Cohen-Macaulay（简称C-M），如果S作为i？-模块等于/的高度。理想f有ap-线性分解，如果f是由p次形式生成的，且f的分次极小分解的所有映射都有自由R-模的线性元。具有线性分解的Cohen-Macaulay代数的例子包括最小重数环[17]，由一般对称矩阵的次极大子式定义的簇的坐标环[13]，由一般矩阵的极大子式定义的簇的坐标环[3]，Sally [ 17]和Schenzel [ 18]分别研究了p - 2情形和一般情形下具有线性分解的Cohen-Macaulay环;在[21，9，4]中研究了具有线性分解的更一般的环。在这项工作中，我们使用了Kustin，米勒和Ulrich [14]介绍的一些技巧，和Vasconcelos [22]研究了Cohen-Macaulay理想的线性分解的Koszul同调，现叙述本文的内容.在§2中，我们考虑高度为g的C-M理想f具有p-线性分解。当g = 2时，Avramov和Herzog [ 1 ]证明了f的Koszul同调是Cohen-Macaulay。我们能够证明，如果/是一般的完全交集，满足g > 3且p> 2，那么/的第一个Koszul同调模不是C-M。在§3中，我们对Cavaliere、Rossi和Valla [2]的结果给出了一个略有不同的证明，该结果用线性分解来表征C-M环;这样
Abstract. The first Koszul homology module of a Cohen-Macaulay ideal witha linear resolution is studied and some new examples of rings of minimal mul-tiplicity are presented. 1. Introduction Let R be a polynomial ring over a field k, and let / be a graded ideal ofR. The algebra S = R/I is Cohen-Macaulay (C-M for short) if the projectivedimension of S as an i?-module is equal to the height of /. The ideal / has ap-linear resolution if / is generated by forms of degree p and if all the mapsof its graded minimal resolution by free .R-modules have linear entries. We saythat S has a p-linear resolution if / does, and that / is C-M if S is.Examples of Cohen-Macaulay algebras with linear resolutions include ringsof minimal multiplicity [17], the coordinate ring of a variety defined by the submaximal minors of a generic symmetric matrix [13], the coordinate ring of a variety defined by the maximal minors of a generic matrix [3], and some facerings [5].Cohen-Macaulay rings with linear resolutions have been studied by Sally [ 17]for the case p — 2, and by Schenzel [ 18] for the general case; more generalrings with linear resolutions have been examined in [21, 9, 4].In this work we use some of the techniques introduced by Kustin, Miller, andUlrich [14], and by Vasconcelos [22] to study the Koszul homology of Cohen-Macaulay ideals with linear resolution.We now describe the contents of this paper. In §2 we consider a C-M ideal/ of height g with a p-linear resolution. If g = 2, Avramov and Herzog [ 1 ]have shown that the Koszul homology of / is Cohen-Macaulay. We are ableto prove that if / is generically a complete intersection satisfying g > 3 andp > 2 then the first Koszul homology module of / is not C-M.In §3 we present a somewhat different proof of a result due to Cavaliere,Rossi, and Valla [2] which characterizes C-M rings with linear resolution; such