Bounds for Betti Numbers

In this paper we prove parts of a conjecture of Herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded kth syzygy module over the polynomial ring. If in addition the module is n-graded we show that the conjecture holds in full generality. Furthermore, we give lower and upper bounds for the graded Betti numbers of graded ideals with a linear resolution and a fixed number of generators.