Perfect closure detects injective dimension

Abstract

Let $R$ be a noetherian local ring of prime characteristic $p$, and let $R^{\infty }$ denote its perfect closure. We prove that a finitely generated \(R\)-module $N$ has finite injective dimension if and only if ${\mathrm{Ext}}_R^i(R^{\infty },N)=0$ for all $i>0$. As a Gorenstein counterpart, we show that finiteness of the Gorenstein injective (or projective) dimension of $R^{\infty }$ forces $R$ to be Gorenstein, and we relate this to a non-noetherian Cohen factorization of $R\to R^{\infty }$. Applications include preservation of Gorensteinness under weakly etale extensions, structural results on F-coherent and weakly F-nilpotent rings, along with the ascent of Frobenius closure of a parameter ideal to its powers. Finally, assuming ${\mathrm{Ext}}_R^i(\frac{R}{\mathfrak{m}},R^{\infty })=0$ for some $i>\dim (R)$, we show $R$ is regular. This has some applications.