Sequences defined by h-vectors

In this paper we consider the sequence whose n^{th} term is the number of h-vectors of length n. We show that the n^{th} term of this sequence is bounded above by the n^{th} Fibonacci number and bounded below by the number if integer partitions of n into distinct parts. Further we show embedded sequences that directly relate to integer partitions

Recursive strategy for decomposing Betti tables of complete intersections

We introduce a recursive decomposition algorithm for the Betti diagram of a complete intersection using the diagram of a complete intersection defined by a subset of the original generators. This alternative algorithm is the main tool that we use to investigate stability and compatibility of the Boij-Soederberg decompositions of related diagrams; indeed, when the biggest generating degree is sufficiently large, the alternative algorithm produces the Boij-Soederberg decomposition. We also provide a detailed analysis of the Boij-Soederberg decomposition for Betti diagrams of codimension four complete intersections where the largest generating degree satisfies the size condition

Super-Stretched and Countable Cohen-Macaulay Type

This dissertation defines what it means for a Cohen-Macaulay ring to to be super-stretched. In particular, a super-stretched Cohen-Macaulay ring of positive dimension has h-vector (1), (1,n), or (1,n,1). It is shown that Cohen-Macaulay rings of graded countable Cohen-Macaulay type are super-stretched. Further, one dimensional standard graded Gorenstein rings of graded countable type are shown to be hypersurfaces; this result is not known in higher dimensions. In Chapter 1, some background material is given along with some preliminary definitions. This chapter defines what it means to be stretched and super-stretched. The chapter ends by showing a couple of scenarios when these two notions coincide. Chapter 2 deals with super-stretched rings that are standard graded. We begin the chapter by exploring the graded category and defining what it means to be graded countable Cohen-Macaulay type. Equivalent characterizations of super-stretched are discovered and it is shown that rings of graded countable Cohen-Macaulay type are super-stretched. The chapter ends by analyzing standard graded rings that are super-stretched with minimal multiplicity. In Chapter 3, we examine what it means for a local ring to be super-stretched. Finally, Chapter 4 uses the previous results to give a partial answer to the following question: Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite Cohen-Macaulay representation type? In particular, it is shown there is a positive answer when the ring is not Gorenstein. Throughout the chapter, many different cases of graded countable Cohen-Macaulay type are classified. Further, the Gorenstein case is studied is shown to be helpful in giving support to the following folklore conjecture: a Gorenstein ring of countable Cohen-Macaulay representation type is a hypersurface. It is shown that the conjecture holds for one dimensional standard graded Cohen-Macaulay rings of graded countable Cohen-Macaulay type

Ideals with Larger Projective Dimension and Regularity

We define a family of homogeneous ideals with large projective dimension and regularity relative to the number of generators and their common degree. This family subsumes and improves upon constructions given in [Cav04] and [McC]. In particular, we describe a family of three-generated homogeneous ideals in arbitrary characteristic whose projective dimension grows asymptotically as sqrt{d}^(sqrt(d) - 1).Comment: 10 pages. This work was completed at the MRC for Commutative Algebra in Snowbird, UT, which was generously supported by the AM