Asymptotic behaviour of certain sets of prime ideals

Abstract:

Let $I_{1}, \ldots ,I_{g}$ be ideals of the commutative ring $R$, let $M$ be a Noetherian $R$-module and let $N$ be a submodule of $M$; also let $A$ be an Artinian $R$-module and let $B$ be a submodule of $A$. It is shown that, whenever $\left (a_{m}\left (1\right ),\ldots ,a_{m}\left (g\right )\right )_{m\in \mathbb {N}}$ is a sequence of $g$-tuples of non-negative integers which is non-decreasing in the sense that $a_{i}\left (j\right )\leq a_{i+1}\left (j\right )$ for all $j=1,\ldots ,g$ and all $i\in \mathbb {N}$, then Ass$_{R}\left (M/ I_{1}^{a_{n}\left (1\right )}\ldots I_{g}^{a_{n}\left (g\right )}N\right )$ is independent of $n$ for all large $n$, and also Att$_{R}\left (B:_{A}I_{1}^{a_{n}\left (1\right )}\ldots I_{g}^{a_{n}\left (g\right )}\right )$ is independent of $n$ for all large $n$. These results are proved without any regularity conditions on the ideals $I_{1}, \ldots ,I_{g}$, and so (a special case of) the first answers in the affirmative a question raised by S. McAdam.