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

Suppose that $M$ is a finitely-generated graded module of codimension $c\geq 3$overapolynomialringandthattheregularityof$M$isatmost$2a-2$where$a\geq 2$istheminimaldegreeofafirstsyzygyof$M$. Then we show that the sum of the betti numbers of $M$ is at least $\beta_0(M)(2^c + 2^{c-1})$\mathrm{.}Inaddition,if$c \geq 9$thenforeach$1\leq i\leq \lceil c/2\rceil$,weshow$\beta_i(M)\geq 2\beta_0(M){c \choose i}$.