Extremal Betti numbers of graded ideals

The possible extremal Betti numbers of graded ideals in the polynomial ring K[x₁,…,x_n] in n variables with coefficients in a field K are studied, completing our results in [7]. In case char(K) = 0 we determine, given any integers r < n, the conditions under which there exists a graded ideal I ? K[x₁,…, x_n] with extremal Betti numbers $\beta_{k_{i}k_{i}+\ell_{i}}\ {\rm for}\ i=1,\cdots,r$ . We also treat a similar problem for squarefree lexsegment ideals.     

Algebraic Shifting and Exterior and Symmetric Algebra Methods

We define and study Cartan–Betti numbers of a graded ideal J in the exterior algebra over an infinite field which include the usual graded Betti numbers of J as a special case. Following ideas of Conca regarding Koszul–Betti numbers over the symmetric algebra, we show that Cartan–Betti numbers increase by passing to the generic initial ideal and the squarefree lexsegement ideal, respectively. Moreover, we characterize the cases where the inequalities become equalities. As combinatorial applications of the first part of this note and some further symmetric algebra methods we establish results about algebraic shifting of simplicial complexes and use them to compare different shifting operations. In particular, we show that each shifting operation does not decrease the number of facets, and that the exterior shifting is the best among the exterior shifting operations in the sense that it increases the number of facets the least.     

t-Spread strongly stable monomial ideals*

We introduce the concept of t-spread monomials and t-spread strongly stable ideals. These concepts are a natural generalization of strongly stable and squarefree strongly stable ideals. For the study of this class of ideals we use the t-fold stretching operator. It is shown that t-spread strongly stable ideals are componentwise linear. Their height, their graded Betti numbers and their generic initial ideal are determined. We also consider the toric rings whose generators come from t-spread principal Borel ideals.     

Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph ℋ appears within the resolution of its edge ideal ℐ(ℋ). We discuss when recursive formulas to compute the graded Betti numbers of ℐ(ℋ) in terms of its sub-hypergraphs can be obtained; these results generalize our previous work (Hà, H.T., Van Tuyl, A. in J. Algebra 309:405–425, 2007) on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are “well behaved.” For such a hypergraph ℋ (and thus, for any simple graph), we give a lower bound for the regularity of ℐ(ℋ) via combinatorial information describing ℋ and an upper bound for the regularity when ℋ=G is a simple graph. We also introduce triangulated hypergraphs that are properly-connected hypergraphs generalizing chordal graphs. When ℋ is a triangulated hypergraph, we explicitly compute the regularity of ℐ(ℋ) and show that the graded Betti numbers of ℐ(ℋ) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs. Dedicated to Anthony V. Geramita on the occasion of his 65th birthday.     

On the Resolution of Path Ideals of Cycles

We give a formula to compute all the top degree graded Betti numbers of the path ideal of a cycle. Also we will find a criterion to determine when Betti numbers of this ideal are nonzero and give a formula to compute its projective dimension and regularity.     

An Improved Multiplicity Conjecture for Codimension 3 Gorenstein Algebras

The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen–Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture in the Gorenstein case.     

On the extremal Betti numbers of the binomial edge ideal of closed graphs

We study the equality of the extremal Betti numbers of the binomial edge ideal and those of its initial ideal for a closed graph G. We prove that in some cases there is a unique extremal Betti number for and as a consequence there is a unique extremal Betti number for and these extremal Betti numbers are equal.     

Maximal Betti numbers

We provide a short proof that the lexicographic ideal has the greatest Betti numbers among all graded ideals with a fixed Hilbert function.

     

A Gorenstein simplicial complex for symmetric minors

We show that the ideal generated by the (n - 2) minors of a general symmetric n by n matrix has an initial ideal that is the Stanley–Reisner ideal of the boundary complex of a simplicial polytope and has the same graded Betti numbers.     

Non-degenerate curves with maximal Hartshorne-Rao module

Extending results for space curves we establish bounds for the cohomology of a non-degenerate curve in projective $n$-space. As a consequence, for any given $n$ we determine all possible pairs $(d, g)$ where $d$ is the degree and $g$ is the (arithmetic) genus of the curve. Furthermore, we show that curves attaining our bounds always exist and describe properties of these extremal curves. In particular, we determine the Hartshorne-Rao module, the generic initial ideal and the graded Betti numbers of an extremal curve. Dedicated to Silvio Greco on the occasion of his 60th birthdayMathematics Subject Classification (2000):14H50, 13D45.