Alexander Duality Theorem and Second Betti Numbers of Stanley–Reisner Rings

HILBERT SCHEMES and MAXIMAL BETTI NUMBERS over VERONESE RINGS

We show that Macaulay’s Theorem, Gotzmann’s Persistence Theorem, and Green’s Theorem hold over a Veronese toric ring R. We also prove that the Hilbert scheme over R is connected; this is an analogue of Hartshorne’s theorem that the Hilbert scheme over a polynomial ring is connected. Furthermore, we prove that each lex ideal in R has the greatest Betti numbers among all graded ideals with the sa...

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...

Dirac's theorem on chordal graphs and Alexander duality

By using Alexander duality on simplicial complexes we give a new and algebraic proof of Dirac’s theorem on chordal graphs.

-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...

Coverings and Betti Numbers