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Properties of local rings and resolutions of modules
Abstract
We study relations between properties of different types of resolutions of modules over a commutative noetherian local ring and properties of the ring. We explain how properties of the ring impose uniform behavior on the resolutions of certain modules, and conversely, how behavior of resolutions, or of invariants, of specific modules imply nice properties of the ring. Let R be a d-dimensional local ring containing a field, m its maximal ideal and x1,..., xd a system of parameters for. If depth R ≥ d - 1 and the local cohomology module [special characters omitted] is finitely generated, then there exists an integer n such that the modules R/[special characters omitted] have the same Betti numbers, for all i ≥ n. A finite R-module M is said to be Gorenstein if Exti(k,M) = 0 for all i ≠ dim R. Assume that R has a contracting endomorphism, that is to say, a homomorphism of rings ϕ: R → R such that ϕi[special characters omitted] for some i ≥ 1. Letting ϕ R denote the R-module R with action induced by ϕ, we prove: A finite R-module M is Gorenstein if and only if HomR(ϕ R,M) ≅ M and [special characters omitted](ϕR,M) = 0 for 1 ≤ i ≤ depth R.
Subject Area
Mathematics
Recommended Citation
Rahmati, Hamidreza, "Properties of local rings and resolutions of modules" (2009). ETD collection for University of Nebraska-Lincoln. AAI3366072.
https://digitalcommons.unl.edu/dissertations/AAI3366072