Betti numbers for modules and primary components of three-generated ideals
Abstract
This thesis consists of two independent chapters. Their titles are joined to form the title of the thesis. We study the non-decreasing and the termwise, alternating and strong exponential growth properties of Betti numbers. Let $(R, \underline{\rm m}, k)$ be a Noetherian local ring and M be a finitely generated R-module of infinite projective dimension. We obtain two inequalities on the sequence $b\sbsp{n}{R}(M)$ of Betti numbers and find a few applications. If R is Artinian with $\underline{\rm m}\sp3 = 0,$ the ideas and proofs of Lescot are analysed to give cases of termwise, alternating and strong exponential growth. If R is Artinian with $\underline{\rm m}\sp4 = 0,$ we obtain new cases of termwise, alternating and strong exponential growth by using techniques of linear algebra. Avramov asked if $b\sbsp{n}{R}(M)$ is eventually non-decreasing for any finitely generated module M over a local ring R. We give a positive answer if one of the Hilbert coefficients $e\sb1, e\sb2$ and $e\sb3$ of R is at least the sum of the other two. The second chapter concerns a three-generated ideal I and its unmixed part J in a regular local ring. A result of Huneke-Ulrich on primary components of three-generated ideals is deduced using elementary methods. By working on finite length modules over regular local rings, we also obtain new results on I: J and the primary components of I.
Degree
Ph.D.
Advisors
Huneke, Purdue University.
Subject Area
Mathematics
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