PreprintarXiv licencearXiv
On Perfectoidization of Finite Algebras over a Perfectoid Ring
Abstract
We study general properties of the perfectoidization of finite algebras over a perfectoid ring, which helps to understand some precise and explicit descriptions. For example, we prove that if A=R[t]/(m(t)) where m(t) is monic, R is perfectoid and the discriminant d of m(t) is a non-zero divisor of R satisfying a bounded torsion condition, then dApfd⊂ A. We also prove a density criterion reducing the construction of the perfectoidization to adjoining suitable p-power roots modulo p. In the second part of the paper, we compute perfectoidizations in several families of examples, including Kummer-type extensions and split finite algebras.
Related
- Null polynomials over a finite ring need not form a two-sided ideal
- Homological Detection by Perfectoid Algebras
- Supercharacter theory and applications to Ramanujan sums over a finite Frobenius ring
- simple cartan-free modules of finite rank over the super-virasoro algebras
- On the trace identities of 2-dimensional tr-prime algebras over a finite field
BibTeX
@misc{ishizuka2026perfectoidization,
title = {{On Perfectoidization of Finite Algebras over a Perfectoid Ring}},
author = {Ryo Ishizuka and Léo Navarro Chafloque},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.12229},
eprint = {2606.12229},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.12229},
}
RIS
TY - UNPB
TI - On Perfectoidization of Finite Algebras over a Perfectoid Ring
AU - Ryo Ishizuka
AU - Léo Navarro Chafloque
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2606.12229
UR - https://arxiv.org/abs/2606.12229
AB - We study general properties of the perfectoidization of finite algebras over a perfectoid ring, which helps to understand some precise and explicit descriptions. For example, we prove that if A=R[t]/(m(t)) where m(t) is monic, R is perfectoid and the discriminant d of m(t) is a non-zero divisor of R satisfying a bounded torsion condition, then dA_pfd⊂ A. We also prove a density criterion reducing the construction of the perfectoidization to adjoining suitable p-power roots modulo p. In the second part of the paper, we compute perfectoidizations in several families of examples, including Kummer-type extensions and split finite algebras.
ER -