论文标题
Ulrich拆分环
Ulrich split rings
论文作者
论文摘要
如果乌尔里希模块分裂的任何简短的精确序列,则局部cohen-macaulay环称为乌尔里希·斯普利特。在本文中,我们启动了Ulrich拆分环的研究。我们证明了该属性的几个必要或足够的标准,将其与残基领域和共同体学的共同体联系起来。我们表征了乌尔里希(Ulrich)的小尺寸拆分环。在复杂数字上,$ 2 $维的Ulrich拆分环(正常且具有最小的多重性)精确地是循环商的奇异点,最多有两个不可兼容的Ulrich模块,直到同构为同构。我们提供了几种方法来构建Ulrich拆分环,并提供一系列应用程序,从最大Cohen--Macaulay模块的测试理想到通过消失$ \ operatatorName {ext} $来检测投影/注射模块。
A local Cohen--Macaulay ring is called Ulrich-split if any short exact sequence of Ulrich modules split. In this paper we initiate the study of Ulrich split rings. We prove several necessary or sufficient criteria for this property, linking it to syzygies of the residue field and cohomology annihilator. We characterize Ulrich split rings of small dimensions. Over complex numbers, $2$-dimensional Ulrich split rings, which are normal and have minimal multiplicity, are precisely cyclic quotient singularities with at most two indecomposable Ulrich modules up to isomorphism. We give several ways to construct Ulrich split rings, and give a range of applications, from test ideal of the family of maximal Cohen--Macaulay modules, to detecting projective/injective modules via vanishing of $\operatorname{Ext}$.