Asymptotic associate primes

Abstract 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 finite subset A of Spec ( A ) such that for every p ∈ A , at least one of the following holds true: (i) p ∈ Ass ( Tor 2 i A ( M , N ) ) for all i ≫ 0 ; (ii) p ∈ Ass ( Tor 2 i + 1 A ( M , N ) ) for all i ≫ 0 . We also analyze the asymptotic behavior of Tor i A ( M , A / I n ) for i , n ≫ 0 in the case when I is principal or I has a principal reduction generated by a regular element.