Rigidity of linear strands and generic initial ideals

Abstract

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 $\beta_{ii+k}^{S}(S/I) = \beta_{ii+k}^{S}(S/\operatorname{Gin}(I))$ for some $i > 1$ and $k \geq 0$, then $\beta_{qq+k}^{S}(S/I) = \beta_{qq+k}^{S}(S/\operatorname{Gin}(I))$ for all $q \geq i$, where $I \subset S$ is a graded ideal. Second, we show that if $\beta_{ii+k}^{E}(E/I) = \beta_{ii+k}^{E}(E/\operatorname{Gin}(I))$ for some $i > 1$ and $k \geq 0$, then $\beta_{qq+k}^{E}(E/I) = \beta_{qq+k}^{E}(E/\operatorname{Gin}(I))$ for all $q \geq 1$, where $I \subset E$ is a graded ideal. In addition, it will be shown that the graded Betti numbers $\beta_{ii+k}^{R}(R/I) = \beta_{ii+k}^{R}(R/\operatorname{Gin}(I))$ for all $i \geq 1$ if and only if $I_{\langle k \rangle}$ and $I_{\langle k+1 \rangle}$ have a linear resolution. Here $I_{\langle d \rangle}$ 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.