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Minimal generating sets of ideals and torsion in the tensor product of I and (,R)I('-1)
Abstract
Let (R, m) be a commutative one-dimensional local Noetherian domain. Assume the integral closure R is a discrete valuation ring and the residue fields of R and R are the same. Let Rx be the maximal ideal of R and let I be a non-principal ideal of R. A question, motivated by the work of Auslander in the early 1960's, is whether the torsion submodule of $I\ \otimes\sb{R}\ I\sp{-1}$ is always non-zero. We investigate this question by examining the stronger condition $\mu\sb{R}(I)\mu\sb{R}(I\sp{-1})>\mu\sb{R}(II\sp{-1})$ (where $\mu$ denotes the number of generators). We will show that if both m and I are generated by powers of x then this strict inequality holds whenever R has maximal embedding dimension, embedding dimension two or multiplicity at most seven. Without the assumption that m and I are generated by powers of x, we will prove the strict inequality holds whenever R has multiplicity at most four.
Subject Area
Mathematics
Recommended Citation
Herzinger, Kurt D, "Minimal generating sets of ideals and torsion in the tensor product of I and (,R)I('-1)" (1996). ETD collection for University of Nebraska-Lincoln. AAI9628235.
https://digitalcommons.unl.edu/dissertations/AAI9628235