We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.

Regularity, Depth and Arithmetic Rank of Bipartite Edge Ideals

In this paper, we prove that the arithmetical rank of a squarefree monomial ideal I is equal to the projective dimension of R/I in the following cases: (a) I is an almost complete intersection; (b) arithdeg?I=reg?I; (c) arithdeg?I=indeg?I+1.

We also classify all almost complete intersection squarefree monomial ideals in terms of hypergraphs, and use this classification in the proof in case (c).

/pdf/arithmetical-rank-of-squarefree-monomial-ideals-of-small-p72cw34t33.pdf

Arithmetical rank of squarefree monomial ideals of small arithmetic degree

In this paper, we prove that the length of a Lyubeznik resolution of a monomial ideal gives an upper bound for the arithmetical rank of the ideal.

/pdf/lyubeznik-resolutions-and-the-arithmetical-rank-of-monomial-55i9z7a5bm.pdf

Lyubeznik resolutions and the arithmetical rank of monomial ideals

HOMOLOGICAL INVARIANTS OF MONOMIAL AND BINOMIAL IDEALS Neelakandhan Manoj Kummini The University of Kansas. Advisor: Craig Huneke. August 2008. In this dissertation, we study numerical invariants of minimal graded free resolutions of homogeneous ideals in a polynomial ring R. Chapters 2, 3 and 4 deal with homological invariants of edge ideals of bipartite graphs. First, in Chapter 2, we relate regularity and depth of bipartite edge ideals to combinatorial invariants of the graphs. Chapter 3 discusses arithmetic rank, and shows that some classes of Cohen-Macaulay bipartite edge ideals define set-theoretic complete intersections. It is known, due to G. Lyubeznik, that arithmetic rank of a square-free monomial ideal I is at least the projective dimension of R/I. As an application of the results in Chapter 2, we show in Chapter 4 that the multiplicity conjectures of J. Herzog, C. Huneke and H. Srinivasan hold for bipartite edge ideals, and that if the conjectured bounds hold with equality, then the ideals are Cohen-Macaulay and has a pure resolution. Chapter 5 describes joint work with G. Caviglia, showing that any upper bound for projective dimension of an ideal supported on N monomials counted with multiplicity is at least 2N/2. We give the example of a binomial ideal, whose projective dimension grows exponentially with respect to the number of monomials appearing in a set of generators. Finally, in Chapter 6, we study Alexander duality, giving an alternate proof of a theorem of K. Yanagawa which states that for a square-free monomial ideal I, R/I has Serre’s property (Si) if and only if its Alexander dual has a linear resolution up to homological

https://kuscholarworks.ku.edu/bitstream/handle/1808/4199/umi-ku-2656_1.pdf?sequence=1

Homological Invariants of Monomial and Binomial Ideals

Let I be a squarefree monomial ideal of a polynomial ring S. In this article, we prove that the arithmetical rank of I is equal to the projective dimension of S/I when one of the following conditions is satisfied: (1) μ(I) ≤5; (2) arithdeg I ≤ 4.

Arithmetical Rank of Squarefree Monomial Ideals Generated by Five Elements or with Arithmetic Degree Four

An etale analog of the Goresky-Macpherson formula for subspace arrangements

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An etale analog of the Goresky-Macpherson formula for subspace arrangements