Asymptotic behaviour of certain sets of associated prime ideals of Ext-modules

Abstract

Let R be a commutative Noetherian ring, $\frak {a}$ be an ideal of R and M be a finitely generated R-module. Melkersson and Schenzel asked whether the set ${\rm Ass}_R{\rm Ext}^i_R(R/\frak {a}^j, M)$ becomes stable for a fixed integer i and sufficiently large j. This paper is concerned with this question. In fact, we prove that if s ≥ 0 and n ≥ 0 such that ${\rm dim}({\rm Supp}_R H^i_\frak {a}(M))\leq s$ for all i with i < n, then $({\rm i}) \quad{\rm the\,set}\,\left(\bigcup_{j > 0}{\rm Supp}_R{\rm Ext}^i_R\left(R/\frak {a}^j, M\right)\right)_{\geq s}$ is finite for all i with i < n, and $({\rm ii})\quad{\rm\,the\,set}\,\left(\bigcup_{j>0}{\rm Ass}_R{\rm Ext}^i_R\left(R/\frak {a}^j, M\right)\right)_{\geq s}$ is finite for all i with i ≤ n, where for a subset T of Spec(R), we set $(T)_{\geq s}\,:=\,\left\{\frak {p} \in T \ | \ {\rm dim}(R/\frak {p})\geq s\right\}$ . Also, among other things, we show that if n ≥ 0, R is semi-local and ${\rm Supp}_{R}H^i_\frak {a}(M)$ is finite for all i with i < n, then $\bigcup_{j > 0}{\rm Ass}_R{\rm Ext}^i_R(R/\frak {a}^j, M)$ is finite for all i with i ≤ n.