In this paper we study simplicial complexes as higher dimensional graphs in order to produce algebraic statements about their facet ideals. We introduce a large class of square-free monomial ideals with Cohen-Macaulay quotients, and a criterion for the Cohen-Macaulayness of facet ideals of simplicial trees. Along the way, we generalize several concepts from… (More)
We generalize the concept of a cycle from graphs to simplicial complexes. We show that a simplicial cycle is either a sequence of facets connected in the shape of a circle, or is a cone over such a structure. We show that a simplicial tree is a connected cycle-free simplicial complex, and use this characterization to produce an algorithm that checks in… (More)
Index 130 Bibliography 133 Introduction This book is based on six lectures and tutorials that were prepared for a workshop in computational commutative algebra at the Harish Chandra Research Institute (HRI) at Allahabad, India in December 2003. The workshop was aimed at graduate students and was conducted as part of the conference on Commutative Algebra and… (More)
Tanisaki introduced generating sets for the defining ideals of the schematic intersections of the closure of conjugacy classes of nilpotent matrices with the set of diagonal matrices. These ideals are naturally labeled by integer partitions. Given such a partition λ, we define several methods to produce a reduced generating set for the associated ideal I λ.… (More)
The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled " Impartial Combinatorial Misère Games " by Meghan Rose Allen in partial fulfillment of the requirements for the degree of Master of Science. Permission is herewith granted to Dalhousie University to circulate and to have… (More)
We present an algorithm that checks in polynomial time whether a simplicial complex is a tree. We also present an efficient algorithm for checking whether a simplicial complex is grafted. These properties have strong algebraic implications for their corresponding facet ideals.
Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebraic properties of the edge ideal from the deck of G, that is, from the collection of subgraphs obtained by removing a vertex from G. These properties include the Krull dimension, the Hilbert function, and all the graded Betti numbers i,j where j < n. We also… (More)
In this paper we give a necessary and sufficient combinatorial condition for a monomial ideal to have a linear resolution over fields of characteristic 2.
We provide a simple method to compute the Betti numbers of the Stanley-Reisner ideal of a simplicial tree and its Alexander dual. Simplicial trees [F1] are a class of flag complexes initially studied for the properties of their facet ideals. In this short note we give a short and straightforward method to compute the Betti numbers of their Stanley-Reisner… (More)