

Title of Thesis
Trees and CohenMacaulay 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, CohenMacaulay,
Monomial, Ideals, chordal, matroid, co dimension 
Abstract In this thesis we
give a structure theorem for CohenMacaulay 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 socalled 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.

