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 $I_1,\dots ,I_t$ ideals of $R$. Let $M$ and $N$ be finitely generated $R$-modules and $N^{\prime }\subseteq N$ a submodule. For every fixed $i\ge 0$, the sets ${\mathrm{Ass}}_R\left({\mathrm{Ext}}_R^i(M,N/I_1^{n_1}\cdots I_t^{n_t}{N}^{\prime })\right)$ and ${\mathrm{Ass}}_R\left({\mathrm{Tor}}_i^R(M,N/I_1^{n_1}\cdots I_t^{n_t}{N}^{\prime })\right)$ are independent of $(n_1,\dots ,n_t)$ for all sufficiently large $n_1,\dots ,n_t$.

3 pages, revised version