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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
Degree Granting Institution:University of Illinois at Urbana-Champaign
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 R-sequence can be divided into two categories according to their betti numbers. Either for all i $\beta\sbsp{i}{R}(M) \geq (\sbsp{i}{d}) + (\sbsp{i-1}{d-1})$ or for all i $\beta\sbsp{i}{R}(M) \geq (\sbsp{i}{d}) + (\sbsp{\ i}{d-1})$. Thus if $\beta\sbsp{i}{R}(M)$ = $(\sbsp{i}{d})$ then M must be R modulo a maximal R-sequence. 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$\sp{d-1}$ where d is the dimension of the ring up to dimension 4.
Issue Date:1990
Rights Information:Copyright 1990 Charalambous, Hara
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9026154
OCLC Identifier:(UMI)AAI9026154

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