Asymptotic associate primes

We investigate three cases regarding asymptotic associate primes. First, assume $(A,\mathfrak{m})$ is an excellent Cohen-Macaulay (CM) non-regular local ring, and $M = \operatorname{Syz}^A_1(L)$ for some maximal CM $A$-module $L$ which is free on the punctured spectrum. Let $I$ be a normal ideal. In this case, we examine when $\mathfrak{m} \notin \operatorname{Ass}(M/I^nM)$ for all $n \gg 0$. We give sufficient evidence to show that this occurs rarely. Next, assume that $(A,\mathfrak{m})$ is excellent Gorenstein non-regular isolated singularity, and $M$ is a CM $A$-module with $\operatorname{projdim}_A(M) = \infty$ and $\dim(M) = \dim(A) -1$. Let $I$ be a normal ideal with analytic spread $l(I) < \dim(A)$. In this case, we investigate when $\mathfrak{m} \notin \operatorname{Ass} \operatorname{Tor}^A_1(M, A/I^n)$ for all $n \gg 0$. We give sufficient evidence to show that this also occurs rarely. Finally, suppose $A$ is a local complete intersection ring. For finitely generated $A$-modules $M$ and $N$, we show that if $\operatorname{Tor}_i^A(M, N) \neq 0$ for some $i > \dim(A)$, then there exists a non-empty finite subset $\mathcal{A}$ of $\operatorname{Spec}(A)$ such that for every $\mathfrak{p} \in \mathcal{A}$, at least one of the following holds true: (i) $\mathfrak{p} \in \operatorname{Ass}\left( \operatorname{Tor}_{2i}^A(M, N) \right)$ for all $i \gg 0$; (ii) $\mathfrak{p} \in \operatorname{Ass}\left( \operatorname{Tor}_{2i+1}^A(M, N) \right)$ for all $i \gg 0$. We also analyze the asymptotic behaviour of $\operatorname{Tor}^A_i(M, A/I^n)$ for $i,n \gg 0$ in the case when $I$ is principal or $I$ has a principal reduction generated by a regular element.