Characterization Of Local Rings Research Articles

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.

Prethick Subcategories of Modules and Characterizations of Local Rings

This article studies characterizing local rings in terms of homological dimensions. The key tool is the notion of a prethick subcategory which we introduce in this article. Our methods recover the theorems of Salarian, Sather-Wagstaff, and Yassemi.

David Rees, FRS 1918–2013

David Rees, FRS 1918–2013

Local rings with zero-dimensional formal fibers

We study Noetherian local rings whose all formal fibers are of dimension zero. Universal catenarity and going-up property of the canonical map to the completion are considered. We present several characterizations of these rings, including a characterization of Weierstrass preparation type. A characterization of local rings with going up property by a strong form of Lichtenbaum–Hartshorne Theorem is obtained. As an application, we give an upper bound for dimension of formal fibers of a large class of algebras over these rings.

Characterization of local rings

Characterization of local rings