Abstract
Let $I\subseteq R=\kk[x_1,\ldots,x_n]$ be a homogeneous ideal generated by forms of degree~$r$. We show here that the shapes of the Betti tables of the ideals $I^d$ stabilize, in the sense that there exists some $D$ such that for all $d\geq D$, $\b_{i,j+rd}(I^d)\neq 0\Leftrightarrow \b_{i,j+rD}(I^D)\neq 0$. We also produce upper bounds for the stabilization index $\text{Stab\,}(I)$. This strengthens the result of Cutkosky, Herzog and Trung that the Castelnuovo-Mumford regularity $\text{reg\,}(I^d)$ is eventually a linear function in $d$.
Citation
Gwyneth Whieldon. "Stabilization of Betti tables." J. Commut. Algebra 6 (1) 113 - 126, SPRING 2014. https://doi.org/10.1216/JCA-2014-6-1-113
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