Asymptotic associate primes

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:223

Article

Ghosh, Dipankar¹  Mallick, Provanjan²  Puthenpurakal, Tony J.² 

[1] Chennai Math Inst, H1,SIPCOT IT Pk, Chennai 603103, Tamil Nadu, India

[2] Indian Inst Technol, Dept Math, Mumbai 400076, Maharashtra, India

关键词: Asymptotic associate primes;    Asymptotic grade;    Associated graded rings and modules;    Local cohomology;    Tor;    Complete intersections;   

DOI  :  10.1016/j.jpaa.2019.01.007

来源: Elsevier

【 摘 要 】

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 is not an element of 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) = infinity no 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 is not an element of 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(1)(A)(M, N) not equal 0 for some i > dim(A), then there exists a non-empty finite subset A of Spec(A) such that for every p is an element of A, at least one of the following holds true: (i) P is an element of Ass Tor(2i)(A)(M, N) for all i >> 0; (ii) P is an element of Ass Tor(2i+1)(A)(M, N) for all i >> 0. We also analyze the asymptotic behavior of Tor(1)(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. (C) 2019 Elsevier B.V. All rights reserved.

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