A short proof of a result of Katz and West

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 $$ \mathrm {Ass}_R(\mathrm {Ext}_R^i(M,N/I_1^{n_1}\cdots I_t^{n_t} N') ) $$ and $$ \mathrm {Ass}_R(\mathrm {Tor}_i^R(M,N/I_1^{n_1}\cdots I_t^{n_t} N') ) $$ are independent of $(n_1,\ldots ,n_t)$ for all sufficiently large $n_1,\ldots ,n_t$.