Jason McCullough

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 by Caviglia (2004) and McCullough (2011). In particular, we describe a family of three-generated homogeneous ideals, in arbitrary characteristic, whose projective dimension grows asymptotically as a power of the degree of the generators. Highlights? 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 by Caviglia (2004) and McCullough (2011). ? In particular, we describe a family of three-generated homogeneous ideals, in arbitrary characteristic, whose projective dimension grows asymptotically as a power of the degree of the generators.

A FAMILY OF IDEALS WITH FEW GENERATORS IN LOW DEGREE AND LARGE PROJECTIVE DIMENSION

Stillman posed a question as to whether the projective dimension of a homogeneous ideal I in a polynomial ring over a field can be bounded by some formula depending only on the number and degrees of the minimal gener- ators of I. More recently, motivated by work on local cohomology modules in characteristic p, Zhang asked more specifically if the projective dimension of I is bounded by the sum of the degrees of the generators. We define a family of homogeneous ideals in a polynomial ring over a field of arbitrary characteristic whose projective dimension grows exponentially if the number and degrees of the generators are allowed to grow linearly. We therefore answer Zhangs ques- tion in the negative and provide a lower bound to any answer to Stillmans question. We also describe some explicit counterexamples to Zhangs question including an ideal generated by 7 quadrics with projective dimension 15.

HYPERGRAPHS AND REGULARITY OF SQUARE-FREE MONOMIAL IDEALS

We define a new combinatorial object, which we call a labeled hypergraph, uniquely associated to any square-free monomial ideal. We prove several upper bounds on the regularity of a square-free monomial ideal in terms of simple combinatorial properties of its labeled hypergraph. We also give specific formulas for the regularity of square-free monomial ideals with certain labeled hypergraphs. Furthermore, we prove results in the case of one-dimensional labeled hypergraphs.

Bounding Projective Dimension

This paper is a survey of progress on Stillman’s Question: Let J be a homogeneous ideal in a standard graded polynomial ring over a field. Is there a bound on the projective dimension of J depending only on the number of elements in a minimal system of homogenoeus generators of J and their degrees (in particular, independent of the number of variables)?

A multiplicity bound for graded rings and a criterion for the Cohen-Macaulay property

Let R be a polynomial ring over a field. We prove an upper bound for the multiplicity of R/I when I is a homogeneous ideal of the form I = J + (F ), where J is a Cohen-Macaulay ideal and F / ∈ J . We show that ideals achieving this upper bound have high depth, and provide a purely numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersections are given.