Perfect closure detects injective dimension

2026/06/29 by Mohsen Asgharzadeh · 1 citation
Mathematics · #math.AC

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Abstract

Let R be a local ring of prime characteristic p, and let R^∞ denote the perfect closure of R. We prove that a finitely generated R-module N has finite injective dimension if and only if Ext_Rⁱ(R^∞, 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.

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• Homological Detection by Perfectoid Algebras

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