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 $$c\ge 3$$ over a polynomial ring and that the regularity of M is at most $$2a-2$$ where $$a\ge 2$$ 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 $$\beta _0(M)(2^c + 2^{c-1})$$ . Additionally, under the same hypothesis on the regularity, we establish the surprising fact that if $$c \ge 9$$ then the first half of the betti numbers are each at least twice the bound predicted by the Buchsbaum-Eisenbud-Horrocks rank conjecture: for $$1\le i \le \frac{c+1}{2}$$ , $$\beta _i(M) \ge 2\beta _0(M){c \atopwithdelims ()i}$$ .