Asymptotic associate primes

TL;DR

This paper studies the long-term behavior of associated primes in various algebraic contexts, revealing that certain asymptotic properties are rare and characterizing the structure of Tor modules over complete intersection rings.

Contribution

It provides new insights into the asymptotic behavior of associated primes and Tor modules in Cohen-Macaulay, Gorenstein, and complete intersection rings, with specific conditions and rare occurrence results.

Findings

Asymptotic associate primes occur rarely in Cohen-Macaulay non-regular rings.

In Gorenstein isolated singularities, the asymptotic behavior of associated primes in Tor modules is also rare.

For complete intersection rings, the structure of Tor modules stabilizes in a predictable way for large indices.

Abstract

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 = \operatorname{Syz}^A_1(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) $ for all $ 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 $\operatorname{projdim}_A(M) = \infty $ 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 $\mathfrak{m} \notin \operatorname{Ass} \operatorname{Tor}^A_1(M, A/I^n)$ for all $n \gg 0$. 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 $ \operatorname{Tor}_i^A(M, N) \neq 0 $ for some $ i > \dim(A) $, then there exists a non-empty finite subset $ \mathcal{A} $ of $ \operatorname{Spec}(A) $ such that for every $ \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) $ for all $ i \gg 0 $; (ii) $ \mathfrak{p} \in \operatorname{Ass}\left( \operatorname{Tor}_{2i+1}^A(M, N) \right) $ for all $ i \gg 0 $. We also analyze the asymptotic behaviour of $\operatorname{Tor}^A_i(M, A/I^n)$ for $i,n \gg 0$ in the case when $I$ is principal or $I$ has a principal reduction generated by a regular element.