We give bounds for the regularity of the local cohomology of Tor k (A, B) in terms of the graded Betti numbers of A and B, under the assumption that dim Tor 1 (A, B) ≤ 1. We apply the results to syzygies, Gröbner bases, products and powers of ideals, and to the relationship of the Rees and Symmetric algebras. For example we show that any homogeneous linearly presented m-primary ideal has some p...

Let S = K[x1, . . . , xn], let A, B be finitely generated graded S-modules, and let m = (x1, . . . , xn) ⊂ S. We give bounds for the regularity of the local cohomology of Tork (A, B) in terms of the graded Betti numbers of A and B, under the assumption that dim Tor1 (A, B) ≤ 1. We apply the results to syzygies, Gröbner bases, products and powers of ideals, and to the relationship of the Rees an...

the Betti cycle class map. If m = 0, then the Hodge conjecture (classical form) implies that clr,0 is surjective. Beilinson ([Be]) once conjectured that cl U r,m is always surjective. It was Jannsen ([J3]) who was the first to find a counterexample, in the case m = 1, where the complex numbers C are used in an essential way. In contrast to this, one expects the surjectivity of clr,m in the case...

The direct three component modified Mannich reaction via condensation of aldehydes, 2-naphthol or 2,7-naphthalendiol and piperidine to generate Betti bases has been carried out over L-proline (20 mol%) with high efficiency under solvent free conditions at 70 OC. Also, the reaction of aromatic aldehyde and 4-hydroxycoumarin in the presence of L-proline (20 mol%) in the ethanol under reflux condi...

We give a leisurely account of the relationship between volume and L2-Betti numbers on closed, aspherical manifolds based on the results in [5] – albeit with a different point of view. This paper grew out of a talk presented at the first colloquium of the Courant Center in Göttingen in October 2007. 1. Review of L-Betti numbers The L-Betti numbers of a closed Riemannian manifold, as introduced ...