Let $(R,m)$ be a Noetherian local ring, $I, J$ two ideals of $R$, and $A$ an Artinian $R$-module. Let $kge 0$ 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 $tle r$ and each sequence $x_1, ldots , x_t$ which is an $A$-cosequence in dimension $>k$, the set $$ig (overset t{igcuplimits_{i=0}} Att_R(0:_A(x_1^{n_1},ldots, x_i^{n_i}))ig )_{ge k}$$ 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 $tle r$ the set $(overset t{igcuplimits_{i=0}}Att_R(Tor_i^R(R/I, (0:_AJ^n))))_{ge k}$ is stable for large $n$.
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机译：假设$（R， m）$是Noetherian局部环，$ I，J $是$ R $的两个理想值，而$ A $是Artinian $ R $模块。假设$ k ge 0 $为整数，而$ r = Width _ {> k}（I，A）$由Nhan-定义的$ I $的维度$> k $中$ A $-余数序列的长度的最大值。 Hoang cite {NhHo}。首先显示出，对于每个$ t le r $和每个序列$ x_1， ldots，x_t $，这是维度$> k $的$ A $余序列，集合$$ big（ overset t { bigcup limits_ {i = 0}} Att_R（0：_A（x_1 ^ {n_1}， ldots，x_i ^ {n_i}）） big）_ { ge k} $$独立于$ n_1， ldots，n_t $。令$ r $为$ Width _ {> k}（0：_AJ ^ n）$的最终值。然后我们的第二个结果说，对于每个$ t le r $，集合$（ overset t { bigcup limits_ {i = 0}} Att_R（ Tor_i ^ R（R / I，（0：_AJ ^ n ））））_ { ge k} $对于较大的$ n $是稳定的。
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