cyclic groups, torsion-free extensions, Betti numbers We describe algorithms to determine extensions of infinite polycyclic groups having certain properties. In particular, we are interested in torsion-free extensions and extensions whose Fitting subgroup has a minimal centre. Then we apply these methods in topological applications. We discuss the calculation of Betti numbers for compact manifo...

Let ∆ be a simplicial complex and I∆ its Stanley–Reisner ideal. We write ∆ for the exterior algebraic shifted complex of ∆ and ∆ for a combinatorial shifted complex of ∆. It will be proved that for all i and j one has the inequalities βii+j(I∆e) ≤ βii+j(I∆c) on the graded Betti numbers of I∆e and I∆c . In addition, the bad behavior of graded Betti numbers of I∆c will be studied.

Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with bounded Betti numbers, which are not eventually periodic. They provide counterexamples to a conjecture of D. Eisenbud, that every module with bounded Betti numbers over a commutative local ring is eventually periodic of period 2 . It is proved however, that the...

Abstract. The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard grad...