Bulletin of the
Korean Mathematical Society BKMS

Let $(R,\m)$ be a Noetherian local ring, $I, J$ two ideals of $R$, and $A$ an Artinian $R$-module. Let $k\ge 0$ be an integer and $r=\Width_{>k}(I,A)$ the supremum of lengths of $A$-cosequences in dimension $>k$ in $I$ defined by Nhan-Hoang \cite{NhHo}. It is first shown that for each $t\le r$ and each sequence $x_1, \ldots , x_t$ which is an $A$-cosequence in dimension $>k$, the set $$\big (\overset t{\bigcup\limits_{i=0}} \Att_R(0:_A(x_1^{n_1},\ldots, x_i^{n_i}))\big )_{\ge k}$$ is independent of the choice of $n_1,\ldots, n_t$. Let $r$ be the eventual value of $\Width_{>k}(0:_AJ^n)$. Then our second result says that for each $t\le r$ the set $(\overset t{\bigcup\limits_{i=0}}\Att_R(\Tor_i^R(R/I, (0:_AJ^n))))_{\ge k}$ is stable for large $n$.

Keywords: asymptotic stability, attached prime ideal, $A$-cosequence in dimension $>k$, width in dimension $> k$

MSC numbers: 13E05, 13E10