A short proof of a result of Katz and West

TL;DR

This paper provides a concise proof of a theorem by Katz and West, demonstrating that associated primes of certain Ext and Tor modules stabilize for large powers of ideals in Noetherian rings.

Contribution

The paper offers a shorter, more elegant proof of a known result on the asymptotic behavior of associated primes in Ext and Tor modules.

Findings

Associated primes of Ext modules stabilize for large ideal powers.

Associated primes of Tor modules stabilize for large ideal powers.

The proof simplifies understanding of asymptotic prime behavior.

Abstract

We give a short proof of a result due to Katz and West: Let $R$ be a Noetherian ring and $I_1,\ldots,I_t$ ideals of $R$. Let $M$ and $N$ be finitely generated $R$-modules and $N' \subseteq N$ a submodule. For every fixed $i \ge 0$, the sets $\mathrm{Ass}_R\left( \mathrm{Ext}_R^i(M, N/I_1^{n_1}\cdots I_t^{n_t} N') \right)$ and $\mathrm{Ass}_R\left( \mathrm{Tor}_i^R(M, N/I_1^{n_1}\cdots I_t^{n_t} N') \right)$ are independent of $(n_1,\ldots,n_t)$ for all sufficiently large $n_1,\ldots,n_t$.