Rigid resolutions and big Betti numbers

In the first part of the paper we answer (positively) a question raised by the first author which has to do with some sort of rigity of the tail of resolution of an ideal. Let $I$ be a homogeneous ideal in a polynomial ring over a field of characteristic 0. Denote by ${\beta }_i(I)$ the $i$-th Betti number of $I$ and by $Gin(I)$ the revlex generic initial ideal of $I$. In general one has ${\beta }_i(I)\le {\beta }_i(Gin(I))$ and we show that if ${\beta }_i(I)={\beta }_i(Gin(I))$ for some $i$ then ${\beta }_j(I)={\beta }_j(Gin(I))$ for all $j>i$. In the second part of the paper we answer a question of Eisenbud and Huneke. We prove that if $I$ is $m$-primary and $I\subset m^d$ then $\beta_i(m^d)\leq \beta_i(Gin(I))$forall$i$.