Rigidity of linear strands and generic initial ideals
Abstract
Description
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 $β_{ii+k}^S(S/I)=β_{ii+k}^S(S/\Gin(I))$ for some $i>1$ and $k \geq 0$, then $β_{qq+k}^S(S/I)= β_{qq+k}^S(S/\Gin(I))$ for all $q \geq i$, where $I\subset S$ is a graded ideal. Second, we show that if $β_{ii+k}^E(E/I)= β_{ii+k}^E(E/\Gin(I))$ for some $i>1$ and $k \geq 0$, then $β_{qq+k}^E(E/I)= β_{qq+k}^E(E/\Gin(I))$ for all $q \geq 1$, where $I\subset E$ is a graded ideal. In addition, it will be shown that the graded Betti numbers $β_{ii+k}^R(R/I)= β_{ii+k}^R(R/\Gin(I))$ 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.
20 pages, the title was changed and some minor corrections were made. To apper Nagoya Mathematical Journal
20 pages, the title was changed and some minor corrections were made. To apper Nagoya Mathematical Journal