ON THE STRUCTURE OF CLOSED IDEALS
ON THE STRUCTURE OF CLOSED IDEALS
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论封闭理想的结构
DOI：
10.7146/math.scand.a-14311
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发表时间：
2001
影响因子：
0.5
通讯作者：
W. Vasconcelos
中科院分区：
文献类型：
--
作者：
J. Brennan;W. Vasconcelos
设R是Notherian环，我们寻求它的一条路映射到它的积分闭包。环将以某种方式指定，例如通过域或Z上的生成元和关系来指定。乘数方法包括以下内容。对于每个包含正则元的理想I，I，HomROI；Iu，c的自同态环可以被标识为R在其全分式环中的整数扩张。我们说I是闭的，如果Ra Homroi；Iu。尽管有算术上的定义，但这一概念将更好地用同源方法来表达。闭理想的例子是主理想，或者更一般的可逆理想，满足Serre条件S2的环的标准模，以及至少两级的理想。当R是积分闭的时，这个条件对R的所有非零理想都是普遍成立的。正是与这最后一个事实有关，我们对非闭理想的兴趣在于：找到R的积分闭包R等于识别足够多的非闭理想I，使得Ra[HomROI；Iu：
Let R be a Noetherian ring for which we seek to map a path to its integral closure. The ring will be specified in some manner, say by generators and relations over a field or over Z. The multiplier approach consists in the following. For each ideal I containing regular elements, the ring of endomorphisms of I, HomROI;IU ,c an be identified as an integral extension of R in its total ring of fractions. We say that I is closed if Ra HomROI;IU. Despite the arithmetical definition, this notion will be better expressed by homological means. Examples of closed ideals are principal ideals, or more generally invertible ideals, canonical modules of rings satisfying Serre’s condition S2 ,a nd ideals of grade at least two. The condition holds universally for all nonzero ideals of R precisely when R is integrally closed. It is in connection with this last fact that our interest in non-closed ideals lies: Finding the integral closure R of R amounts to identifying sufficiently many non-closed ideals I such that Ra [ HomROI;IU: