The eventual shape of Betti tables of powers of ideals

Abstract

Let $G$ be a finitely generated abelian group, and let $S=A[x_1,...,x_n]$ be a $G$-graded polynomial ring over a commutative ring $A$. Let $I_1,...,I_s$ be $G$-homogeneous ideals in $S$, and let $M$ be a finitely generated $G$-graded $S$-module. We show that, when $A$ is Noetherian, the nonzero $G$-graded Betti numbers of $M{I}_1^{t_1}...I_s^{t_s}$ exhibit an asymptotic linear behavior as the $t_i$s get large.

arXiv comment15 pages; To appear in Math. Res. Letters

References in corpus (2)

• Stabilization of the Regularity of Powers of An Ideal
• Stabilization of Betti Tables