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Title of Thesis

Trees and Cohen-Macaulay Monomial Ideals

Author(s)

MUHAMMAD NAEEM

Institute/University/Department Details
Abdus Salam School Of Mathematical Sciences / GC University, Lahore
Session
2010
Subject
Mathematics
Number of Pages
58
Keywords (Extracted from title, table of contents and abstract of thesis)
Trees, Cohen-Macaulay, Monomial, Ideals, chordal, matroid, co dimension

Abstract
In this thesis we give a structure theorem for Cohen-Macaulay monomial ideals of codimension 2, and describe all possible relation matrices of such ideals. We also study the set T (I) of all relation trees of a Cohen–Macaulay monomial ideal of codimension 2. We show that T (I) is the set of bases of a matroid. In case that the ideal has a linear resolution, the relation matrices can be identified with the spanning trees of a connected chordal graph with the property that each distinct pair of maximal cliques of the graph has at most one vertex in common.
We give the equivalent conditions for a squarefree monomial ideal to be a complete intersection. Then we study the set of Cohen–Macaulay monomial ideals with a given radical. Among this set of ideals are the so-called Cohen–Macaulay modifications. Not all Cohen–Macaulay squarefree monomial ideals admit nontrivial Cohen–Macaulay modifications. It is shown that if there exists one such modification, then there exist indeed infinitely many. We also present classes of Cohen–Macaulay square free monomial ideals with infinitely many nontrivial Cohen–Macaulay modifications.

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S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

 
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2

1

BASIC FACTS ABOUT GRAPHS AND MONOMIAL ALGEBRAS

1.1 Monomial ideals

1.2 Cohen-Macaulay modules and rings

1.3 Free resolutions

1.4 Graphs and trees

1.5 Simplicial complexes

1.6 Chordal Graph

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3 2 COHEN-MACAULAY MONOMIAL IDEALS OF CODIMENSION 2

2.1 On the structure of Cohen-Macaulay monomial ideals of codimension 2

2.2 The possible sets of relation trees attached to Cohen-Macaulay monomial ideals of codimension 2

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4 3 COHEN–MACAULAY MONOMIAL IDEALS WITH GIVEN RADICAL

3.1 A Characterization of Monomial Complete Intersections

3.2 On the existence of nontrivial Cohen–Macaulay modifications of squarefree monomial ideals

3.3 Classes of simplicial complexes which admit nontrivial Cohen–Macaualy modifications

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5 4 BIBLIOGRAPHY

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