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

Adam Boocher, Derrick Wigglesworth

• 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}.