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Title:  On lower bounds for the betti numbers of finite length modules 
Author(s):  Charalambous, Hara 
Doctoral Committee Chair(s):  Griffith, Phillip A. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  In this manuscript we consider multigraded modules. Chapter 1 gives the necessary definitions and examples that develop the theory of multigraded modules. A multigraded module has a multigraded minimal resolution. We give the necessary conditions for a matrix to correspond to a multigraded map and show that all associated primes of a multigraded module are multigraded. In Chapter 2 we show that the betti numbers of multigraded modules satisfy the bound: $\beta\sbsp{i}{R}(M) \geq (\sbsp{i}{d})$. The multigraded modules that are not isomorphic to R modulo an Rsequence can be divided into two categories according to their betti numbers. Either for all i $\beta\sbsp{i}{R}(M) \geq (\sbsp{i}{d}) + (\sbsp{i1}{d1})$ or for all i $\beta\sbsp{i}{R}(M) \geq (\sbsp{i}{d}) + (\sbsp{\ i}{d1})$. Thus if $\beta\sbsp{i}{R}(M)$ = $(\sbsp{i}{d})$ then M must be R modulo a maximal Rsequence. In case M is an almost complete intersection we derive its betti numbers. In Chapter 3 we give better bounds for the betti numbers of cyclic multigraded modules using a deformation argument combined with localization. Lastly we point out that the sum of the betti numbers for all modules of finite length is greater than or equal to 2$\sp{d}$ + 2$\sp{d1}$ where d is the dimension of the ring up to dimension 4. 
Issue Date:  1990 
Type:  Text 
Language:  English 
URI:  http://hdl.handle.net/2142/19766 
Rights Information:  Copyright 1990 Charalambous, Hara 
Date Available in IDEALS:  20110507 
Identifier in Online Catalog:  AAI9026154 
OCLC Identifier:  (UMI)AAI9026154 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics