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 $M={\mathrm{Syz}}_1^A(L)$ 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) $forall$ 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 ${\mathrm{projdim}}_A(M)=\mathrm{\infty }$ and $ \dim(M) = \dim(A) -1 $\mathrm{.}Let$ I $beanormalidealwithanalyticspread$ l(I) < \dim(A) $. In this case, we investigate when $\mathfrak{m} \notin \operatorname{Ass} \operatorname{Tor}^A_1(M, A/I^n)$forall$n \gg 0$. 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 ${\mathrm{Tor}}_i^A(M,N)\mathrm{\ne }0$ for some $ i

\dim(A) $,thenthereexistsanon-emptyfinitesubset$ $\mathcal{A}$ $of$ \operatorname{Spec}(A) $suchthatforevery$ \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) $forall$ i \gg 0 $;(ii)$ \mathfrak{p} \in \operatorname{Ass}\left( \operatorname{Tor}{2i+1}^A(M, N) \right) $forall$ i \gg 0 $. We also analyze the asymptotic behaviour of ${\mathrm{Tor}}_i^A(M,A\mathrm{/}{I}^n)$ for $i,n \gg 0$inthecasewhen$I$isprincipalor$I$ has a principal reduction generated by a regular element.