Homological Detection by Perfectoid Algebras

Mohsen Asgharzadeh; Ryo Ishizuka

Homological Detection by Perfectoid Algebras

Commutative Algebra 2026-07-20 v1 Algebraic Geometry

Authors: Mohsen Asgharzadeh , Ryo Ishizuka

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.

Keywords

perfectoid algebra projective dimension injective dimension Cohen-Macaulay module Gorenstein ring regular local ring mixed characteristic

Cite

@article{arxiv.2607.17801,
  title  = {Homological Detection by Perfectoid Algebras},
  author = {Mohsen Asgharzadeh and Ryo Ishizuka},
  journal= {arXiv preprint arXiv:2607.17801},
  year   = {2026}
}

Comments

Some special cases of the present results appeared in "Perfect Closure Detects Injective Dimension". The current project, however, has a broader mixed-characteristic scope. Due to substantial changes in content and structure, we present this as a separate paper