Homological Detection by Perfectoid Algebras

2026/07/20 by Mohsen Asgharzadeh, Ryo Ishizuka
#math.AC #math.AG

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Abstract

We first establish estimates for the projective and injective dimensions for quotients of perfectoid algebras by radical ideals. As applications, we obtain homological characterizations of modules of finite injective dimension, (big) Cohen--Macaulay modules, Gorenstein local rings, and regular local rings in mixed characteristic in terms of perfectoid algebras. These are mixed characteristic analogues of the corresponding characterizations via Frobenius morphisms in positive characteristic.

Citations

• Perfect closure detects injective dimension

Related

• Homological aspects of perfect algebras
• Perfectoid spaces and the homological conjectures
• A variant of perfectoid Abhyankar's lemma and almost Cohen-Macaulay algebras
• Singularities in mixed characteristic. The perfectoid approach
• Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras