This paper gives a sharp upper bound for the Betti numbers of a finitely generated multigraded R-module, where R = k[x1, . . . , xm] is the polynomial ring over a field k in m variables. The bound is given in terms of the rank and the first two Betti numbers of the module. An example is given which achieves these bounds simultaneously in each homological degree. Using Alexander duality, a bound...

A equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring. By applying the Grothedieck construction one obtains the ring K̂, elements of which, modulo a complex torus of dimension the sum of the odd Betti n...

We prove a tight lower bound on the algebraic Betti numbers of tree and forest ideals and an upper bound on certain graded Betti numbers of squarefree monomial ideals.

This paper defines new intersection homology groups. The basic idea is this. Ordinary homology is locally trivial. Intersection homology is not. It may have significant local cycles. A localglobal cycle is defined to be a family of such local cycles that is, at the same time, a global cycle. The motivating problem is the numerical characterisation of the flag vectors of convex polytopes. Centra...

Let ∆ be a simplicial complex and I∆ its Stanley–Reisner ideal. It has been conjectured that, for each i and j, the graded Betti number βii+j(I∆) of I∆ is smaller than or equal to that of I∆c , where ∆ c is a combinatorial shifted complex of ∆. In the present paper the conjecture will be proved affirmatively. In particular the inequalities βii+j(I∆) ≤ βii+j(I∆lex) hold for all i and j, where ∆ ...

Suppose that f defines a singular, complex affine hypersurface. If the critical locus of f is onedimensional at the origin, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber, Ff,0, of f at the origin, with either integral or Z/pZ coefficients. If the critical locus of f has arbitrary dimension, we show that the smallest possibly non-zero reduced Betti number o...

We describe the cone of Betti tables of all finitely generated graded modules over the homogeneous coordinate ring of three non-collinear points in the projective plane. We also describe the cone of Betti tables of all finite length modules.

In this paper we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first six 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 grou...