Pure resolutions, linear codes, and Betti numbers

Betti numbers of Stanley–Reisner rings with pure resolutions

Let ∆ be simplicial complex and let k[∆] denote the Stanley– Reisner ring corresponding to ∆. Suppose that k[∆] has a pure free resolution. Then we describe the Betti numbers and the Hilbert– Samuel multiplicity of k[∆] in terms of the h–vector of ∆. As an application, we derive a linear equation system and some inequalities for the components of the h–vector of the clique complex of an arbitra...

Betti Numbers and Shifts in Minimal Graded Free Resolutions

Let S = K[x1, . . . ,xn] be a polynomial ring and R = S/I where I ⊂ S is a graded ideal. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan states that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S as well as bounded below by a function of the minimal shifts if R is Cohen–Macaulay. In this paper we study t...

-betti Numbers

The Atiyah conjecture predicts that the L-Betti numbers of a finite CW -complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for positive 1-relator g...

Lascoux-style Resolutions and the Betti Numbers of Matching and Chessboard Complexes

This paper generalizes work of Lascoux and Jo zeeak-Pragacz-Weyman computing the characteristic zero Betti numbers in minimal free resolutions of ideals generated by 2 2 minors of generic matrices and generic symmetric matrices, respectively. In the case of 2 2 minors, the quotients of certain polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we co...

Coverings and Betti Numbers