Complexity and rigidity of Ulrich modules, and some applications

Abstract

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.

arXiv comment22 pages. Accepted at Mathematica Scandinavica

References in corpus (6)

• Homological dimensions of rigid modules
• On the ideal case of a conjecture of Huneke and Wiegand
• Auslander-Reiten conjecture and finite injective dimension of Hom
• Some criteria for regular and Gorenstein local rings via syzygy modules
• Vanishing of (co)homology over deformations of Cohen-Macaulay local rings of minimal multiplicity
• Vanishing of (co)homology of Burch and related submodules

Cited by in corpus (5)

• Auslander-Reiten conjecture and finite injective dimension of Hom
• Homological dimensions of Burch ideals, submodules and quotients
• Auslander-Reiten annihilators
• Integrally closed $\mathfrak{m}$-primary ideals have extremal resolutions
• On modules whose dual is of finite Gorenstein dimension