Perfect closure detects injective dimension

Abstract

Let $R$ be a local ring of prime characteristic $p$, and let $R^\infty$ denote the perfect closure of $R$. We prove that a finitely generated $R$-module $N$ has finite injective dimension if and only if $\operatorname{Ext}_R^i(R^\infty, N) = 0$ for all $i > 0$. This provides a single test module that detects finite injective dimension, thereby refining a classical theorem of Herzog which requires infinitely many Frobenius twist modules ${}^e R$. Analogously, we present the corresponding Tor-side.