Journal of the
Korean Mathematical Society JKMS

Let $R$ be a commutative Noetherian (not necessarily local) ring, $I$ an ideal of $R$ and $M$ a finitely generated $R$-module. In this paper, by computing the local cohomology modules and Ext-modules via the injective resolution of $M$, we proved that, if for an integer $t>0$, ${\dim}_RH_{I}^i(M)\leq k$ for $\forall i0$. This shows that $\bigcup_{n>0}({\text{Ass}}_R{\text{Ext}}_R^i(R/I^n,M))_{\geq k}$ is a finite set for $\forall \, i\leq t$. Also, we prove that $$\scalebox{0.98}{$\displaystyle\bigcup_{i=1}^{r}({\text{Ass}}_RM/(x_1^{n_1}, x_2^{n_2}, \ldots, x_{i}^{n_{i}})M)_{\geq k}\!=\!\bigcup_{i=1}^{r} ({\text{Ass}}_RM/(x_1, x_2, \ldots, x_{i})M)_{\geq k}$}$$ if $x_1, x_2, \ldots, x_r$ is $M$-sequences in dimension $>k$ and $n_1,n_2,\ldots, n_r$ are some positive integers. Here, for a subset $T$ of ${\text{Spec}}(R),$ set $T_{\geq i}=\{{\mathfrak{p}} \in T \mid {\dim}R/{\mathfrak{p}}\geq i\, \}$.

Keywords: local cohomology modules, associated primes, $M$-sequences in dimension $>k$

MSC numbers: 13D45, 13C15