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Let $(R,\frak{m})$ be a Noetherian local ring, $I$ an ideal of $R$ and $N$ afinitely generated $R$-module. Let $k{\ge}-1$ be an integer and $r=\depth_k(I,N)$ the length of a maximal $N$-sequence in dimension $>k$ in $I$defined by M. Brodmann and L. T. Nhan ({Comm. Algebra, 36 (2008), 1527-1536).For a subset $S\subseteq \Spec R$ we set $S_{{\ge}k}={\p\inS\mid\dim(R/\p){\ge}k}$. We first prove in this paper that$\Ass_R(H^j_I(N))_{\ge k}$ is a finite set for all $j{\le}r$}. Let$\fN=\oplus_{n\ge 0}N_n$ be a finitely generated graded $\fR$-module, where$\fR$ is a finitely generated standard graded algebra over $R_0=R$. Let $r$ bethe eventual value of $\depth_k(I,N_n)$. Then our second result says that forall $l{\le}r$ the sets $\bigcup_{j{\le}l}\Ass_R(H^j_I(N_n))_{{\ge}k}$ arestable for large $n$.

On the finiteness and stability of certain sets of associated primes ideals of local cohomology modules