Perfect closure detects injective dimension

Let $R$ be a local ring of prime characteristic $p$, and let $R^{\mathrm{\infty }}$ denote the perfect closure of $R$. We prove that a finitely generated $R$-module $N$ has finite injective dimension if and only if ${\mathrm{Ext}}_R^i(R^{\mathrm{\infty }},N)=0$ 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 ${}^{e}R$. Some applications are given.