Rigidity of linear strands and generic initial ideals

Details Rigidity of linear strands and generic initial ideals Rigidity of linear strands and generic initial ideals

2006-08-25

arxiv.org/abs/math/0608628v2

Mathematics

Commutative Algebra

20 pages, the title was changed and some minor corrections were made. To apper Nagoya Mathematical Journal

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Rigidity of linear strands and generic initial ideals does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Rigidity of linear strands and generic initial ideals, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Rigidity of linear strands and generic initial ideals will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-111884