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=Syz1A(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/InM) 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 projdimA(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∉AssTor1A(M,A/In) 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 ToriA(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(Tor2iA(M,N)) for all i≫0; (ii) p∈Ass(Tor2i+1A(M,N)) for all i≫0. We also analyze the asymptotic behavior of ToriA(M,A/In) for i,n≫0 in the case when I is principal or I has a principal reduction generated by a regular element.