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We investigate three cases regarding asymptotic associate primes. First, assume (A, m) is an excellent Cohen–Macaulay (CM) non-regular local ring, and M= 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 m∉ Ass (M/I n M) for all n≫ 0. We give sufficient evidence to show that this occurs rarely. Next, assume that (A, m) is excellent Gorenstein non-regular isolated singularity, and M is a CM A-module with projdim A (M)=∞ 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 m∉ Ass Tor 1 A (M, A/I n) for all n≫ 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 Tor i A (M, N)≠ 0 for some i> dim⁡(A), then there exists a non-empty …