Let (R,m) be a Noetherian local ring, I an ideal of R and N a
finitely generated R-module. Let kβ₯β1 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\in
S\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 jβ€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 R0β=R. Let r be
the eventual value of \depth_k(I,N_n). Then our second result says that for
all lβ€r the sets \bigcup_{j{\le}l}\Ass_R(H^j_I(N_n))_{{\ge}k} are
stable for large n.Comment: To appear in Communication in Algebr