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)
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 detail here the sparse variant of the algorithm sketched in  for checking if a simplicial complex is a tree. A full worst case complexity analysis is given and several optimizations are discussed. The practical complexity is discussed for some examples.
2001 iv Acknowledgments I'd like to thank my advisor, Ira Gessel, for his unwavering support. He has been all that I ever could have asked for in an advisor, providing invaluable mathematical direction and allowing me the flexibility to deal with the rest of my life. Sara Billey and Susan Parker have not only agreed to sit on my committee, but have kept… (More)