Perfect closure detects injective dimension

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.