Rigid modules and homological dimensions
Victor H. Jorge-P\'erez, Cleto B. Miranda-Neto
TL;DR
This paper explores the relationship between cohomology vanishing and homological dimensions using rigid modules, providing new characterizations of Gorenstein and regular local rings relevant to algebraic geometry.
Contribution
It introduces novel criteria involving Tor-rigid and rigid-test modules to characterize Gorenstein and regular local rings, linking cohomology vanishing to homological dimensions.
Findings
Criteria for module freeness established
New characterizations of Gorenstein rings provided
Connections between cohomology vanishing and homological dimensions demonstrated
Abstract
We make use of the concepts of Tor-rigid and rigid-test modules, among others, to investigate the interplay between cohomology vanishing and the finiteness of several homological dimensions such as projective, injective and Gorenstein injective dimensions. Besides criteria for the freeness of modules, the applications include new characterizations of Gorenstein and regular local rings, which are also of interest in the realm of algebraic geometry.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Commutative Algebra and Its Applications · Homotopy and Cohomology in Algebraic Topology