Ulrich模的复杂性和刚性及其应用

Pub Date : 2022-01-04 DOI:10.7146/math.scand.a-136499

Souvik Dey, D. Ghosh

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引用次数: 6

摘要

我们分析了Ulrich模块是否可以作为检测模块有限同调维数的测试模块，而不一定是极大CM (Cohen-Macaulay)。证明了CM局部环上的Ulrich模具有最大的复杂度和曲率。利用Ulrich模给出了局部环的各种新的表征。我们证明了局部环上每一个维数$s$的Ulrich模都是$(s+1)$-Tor-rigid-test，而不是一般的$s$ -Tor-rigid(其中$s\ge 1$)。在最小复数CM局部环的变形上，我们也研究了其刚性。

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Complexity and rigidity of Ulrich modules, and some applications

We analyze whether Ulrich modules, not necessarily maximal CM (Cohen-Macaulay), can be used as test modules, which detect finite homological dimensions of modules. We prove that Ulrich modules over CM local rings have maximal complexity and curvature. Various new characterizations of local rings are provided in terms of Ulrich modules. We show that every Ulrich module of dimension $s$ over a local ring is $(s+1)$-Tor-rigid-test, but not $s$−Tor-rigid in general (where $s\ge 1$). Over a deformation of a CM local ring of minimal multiplicity, we also study Tor rigidity.

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