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 $\operatorname{Ext}_R^i(R^\infty, N) = 0$ 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 $\operatorname{Ext}^{i}_{R}(\frac{R}{\mathfrak m},R^{\infty})=0$ for some $i>\dim(R)$, we show $R$ is regular. This has some applications.