Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in S and E when passing to their generic initial ideals. First, we prove that if the graded Betti numbers β ii+k(S/I) = β S ii+k (

We prove that the lower bounds for Betti numbers of the rack, quandle and degeneracy cohomol-ogy given in [CJKS] are in fact equalities. We compute as well the Betti numbers of the twisted cohomology introduced in [CES]. We also give a group-theoretical interpretation of the second cohomology group for racks.

We prove that the lower bounds for Betti numbers of the rack, quandle and degeneracy cohomol-ogy given in [CJKS] are in fact equalities. We compute as well the Betti numbers of the twisted cohomology introduced in [CES]. We also give a group-theoretical interpretation of the second cohomology group for racks.

Sums of Betti numbers in arbitrary characteristic Nicholas M. Katz Introduction In [Mil], Milnor gave an explicit upper bound for the sum of the Betti numbers of a complex affine algebraic variety V. If V is defined in ^N, N ≥ 1, by r ≥ 1 equations Fi, i =1 to r, all of degree ≤ d, Milnor showed ‡i h i(V, $) ≤ d(2d-1)2N-1. Oleinik [Ol] and Thom [Th] gave similar results. It is standard (cf. the...