Rigidity of linear strands and generic initial ideals

Let $K$ be a field, $S$ a polynomial ring and $E$ an exterior algebra over $K$, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in $S$ and $E$ when passing to their generic initial ideals. First, we prove that if the graded Betti numbers $\beta_{ii+k}^S(S/I)=\beta_{ii+k}^S(S/\Gin(I))$ for some $i>1$ and $k \geq 0$, then $\beta_{qq+k}^S(S/I)= \beta_{qq+k}^S(S/\Gin(I))$ for all $q \geq i$, where $I\subset S$ is a graded ideal. Second, we show that if $\beta_{ii+k}^E(E/I)= \beta_{ii+k}^E(E/\Gin(I))$ for some $i>1$ and $k \geq 0$, then $\beta_{qq+k}^E(E/I)= \beta_{qq+k}^E(E/\Gin(I))$ for all $q \geq 1$, where $I\subset E$ is a graded ideal. In addition, it will be shown that the graded Betti numbers $\beta_{ii+k}^R(R/I)= \beta_{ii+k}^R(R/\Gin(I))$ for all $i \geq 1$ if and only if $I_{< k >}$ and $I_{< k+1 >}$ have a linear resolution. Here $I_{< d >}$ is the ideal generated by all homogeneous elements in $I$ of degree $d$, and $R$ can be either the polynomial ring or the exterior algebra.