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On perfectoid ideals
Abstract
In this paper, we study perfectoid ideals using the reduction modulo p of perfectoid rings. We show that perfectoid ideals correspond to ideals in the reduction modulo p on which Frobenius is surjective. We establish the existence of universal p-adic power series whose evaluations, together with their compatible p-power roots, generate perfectoidizations of ideals. In the special case of the perfectoidization of ℤpcycl⟨ t1/p∞⟩/(t−1), we describe a linear algebra algorithm for computing finite approximations of such generators. We illustrate this by an explicit approximation modulo $8$ of an element σ∈ (t−1)perfd⊆ ℤ2cycl⟨ t1/2∞⟩ which admits compatible sequences of $2$-power roots and satisfies σ≡ t−1pmod2.
Related
BibTeX
@misc{chafloque2026perfectoid,
title = {{On perfectoid ideals}},
author = {Léo Navarro Chafloque},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31964},
eprint = {2610.31964},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31964},
}
RIS
TY - UNPB
TI - On perfectoid ideals
AU - Léo Navarro Chafloque
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31964
UR - https://arxiv.org/abs/2610.31964
AB - In this paper, we study perfectoid ideals using the reduction modulo p of perfectoid rings. We show that perfectoid ideals correspond to ideals in the reduction modulo p on which Frobenius is surjective. We establish the existence of universal p-adic power series whose evaluations, together with their compatible p-power roots, generate perfectoidizations of ideals. In the special case of the perfectoidization of ℤ_p^cycl⟨ t^1/p^∞⟩/(t−1), we describe a linear algebra algorithm for computing finite approximations of such generators. We illustrate this by an explicit approximation modulo $8$ of an element σ∈ (t−1)_perfd⊆ ℤ_2^cycl⟨ t^1/2^∞⟩ which admits compatible sequences of $2$-power roots and satisfies σ≡ t−1pmod2.
ER -