Large lower bounds for the betti numbers of graded modules with low regularity

Abstract

Suppose that M is a finitely-generated graded module (generated in degree 0) of codimension c ≥ 3 over a polynomial ring and that the regularity of M is at most 2a − 2 where a ≥ 2 is the minimal degree of a first syzygy of M . Then we show that the sum of the betti numbers of M is at least β0(M )(2 c + 2 c−1 ). Additionally, under the same hypothesis on the regularity, we establish the surprising fact that if c ≥ 9 then the first half of the betti numbers are each at least twice the bound predicted by the Buchsbaum-Eisenbud-Horrocks rank conjecture: for 1 ≤ i ≤ c+1 2 , βi(M ) ≥ 2β0(M ) c i .