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PreprintCC BY 4.0arXiv

Homological Detection by Perfectoid Algebras

  • Mohsen Asgharzadeh
  • Ryo Ishizuka
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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.

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Cite

BibTeXRIS
BibTeX
@misc{asgharzadeh2026homological,
  title = {{Homological Detection by Perfectoid Algebras}},
  author = {Mohsen Asgharzadeh and Ryo Ishizuka},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.17801},
  eprint = {2607.17801},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.17801},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Homological Detection by Perfectoid Algebras
AU  - Mohsen Asgharzadeh
AU  - Ryo Ishizuka
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2607.17801
UR  - https://arxiv.org/abs/2607.17801
AB  - 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.
ER  -