A linear function associated to asymptotic prime divisors

Abstract:

Let $R$ be a Noetherian standard ${\mathbb {N}}^{\thinspace d}$-graded ring and $M,N$ finitely generated, ${\mathbb {N}}^{\thinspace d}$-graded $R$-modules. Let $I_{1}, \ldots , I_{s}$ be finitely many homogeneous ideals of $R$. We show that there exist linear functions $f,g : \mathbb {N}^{s} \to \mathbb {N}^{d}$ such that the associated primes over $R_{0}$ of $[\operatorname {Ext}^{i}(N,M/I_{1}^{n_{1}}\cdots I_{s}^{n_{s}}M)]_{m}$ and $[\operatorname {Tor}_{i}(N,M/I_{1}^{n_{1}}\cdots I_{s}^{n_{s}}M)]_{m}$ are stable whenever $m\in {\mathbb {N}}^{\thinspace d}$ satisfies $m\geq f(n_{1},\ldots ,n_{s})$ and $m\geq g(n_{1},\ldots , n_{s})$, respectively.