Mordechai Katzman

Characteristic-independence of Betti numbers of graph ideals

In this paper, we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first 6 Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables n is at most 10, all Betti numbers are independent of the ground field. For n = 11, there exists precisely 4 examples in which the Betti numbers depend on the ground field. This is equivalent to the statement that the homology of flag complexes with at most 10 vertices is torsion free and that there exists precisely 4 non-isomorphic flag complexes with 11 vertices whose homology has torsion.In each of the 4 examples mentioned above the 8th Betti numbers depend on the ground field and so we conclude that the highest Betti number which is always independent of the ground field is either 6 or 7; if the former is true then we show that there must exist a graph with 12 vertices whose 7th Betti number depends on the ground field.

An example of an infinite set of associated primes of a local cohomology module

Let

Parameter-test-ideals of Cohen–Macaulay rings

(R,m)

Associated primes of graded components of local cohomology modules

be a local Noetherian ring, let

THE HILBERT SERIES OF ALGEBRAS OF THE VERONESE TYPE

I\subset R

Rings of Frobenius operators

be any ideal and let

A non-finitely generated algebra of Frobenius maps

M

Some properties of top graded local cohomology modules

be a finitely generated

An algorithm for computing compatibly Frobenius split subvarieties

R

Castelnuovo-Mumford regularity and the discreteness of F -jumping coefficients in graded rings

-module. In 1990 Craig Huneke conjectured that the local cohomology modules