Skip to content
DraconiticResearch
Search Sources API
203,948 studies31 sourcesUpdated 25 min ago
PreprintCC BY-NC-ND 4.0arXiv

Topological transcendence degree and the dimension problem for perfectoid spaces

  • Dimitri Dine
  • Jack J Garzella
Open at sourcePDFDOICite

Abstract

Let K be a perfectoid field of characteristic p with countable value group (for example, K=β„‚pβ™­). We study Zariski-closed subsets ZβŠ† 𝔹Kn,perfdof the n-dimensional perfectoid closed unit ball 𝔹Kn,perfd over K. We use an example of Gleason, and its generalization obtained in the authors' previous work, to show that two natural notions of dimension for such Zariski-closed subsets, the Krull dimension dim(Z) and Huber's iterated transcendence degree dim.tr(Z), do not coincide, and that the gap between them can be arbitrarily large.

Related

  1. The CR Nirenberg Problem, Part II: Existence and Topological Degree
    Jiechen Qiang, Zhongwei Tang, Heming Wang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  2. Automorphic Transcendence Degree, Growth, and Skew Normalization
    Dinh Van Hoang, Vo Ngoc Thieu, Phan Thanh Toan
    arXivPreprintCC BY-NC-ND 4.0Full textAlgebra & number theoryarXiv
  3. Poset dimension and maximum comparability degree
    Carla Groenland, Richard Montgomery, Rajko Nenadov +1
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  4. Explicit M-polynomial and degree-based topological indices of Hanoi graphs
    El-Mehdi Mehiri
    Discrete Applied MathematicsPeer-reviewedLicence not statedFull text2 citedTheoretical & computational chemistryCombinatorics & discrete mathematicsHAL
  5. Entropy measures of degree-based topological indices for hyper carbon nanotubes
    Muhammad Faisal Nadeem, Ayesha Shabbir, Mohamad Nazri Husin +1
    Scientific ReportsPeer-reviewedCC BY-NC-ND 4.0Full textGeometry & topologyCrossref

Cite

BibTeXRIS
BibTeX
@misc{dine2026topological,
  title = {{Topological transcendence degree and the dimension problem for perfectoid spaces}},
  author = {Dimitri Dine and Jack J Garzella},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01418},
  eprint = {2611.01418},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01418},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Topological transcendence degree and the dimension problem for perfectoid spaces
AU  - Dimitri Dine
AU  - Jack J Garzella
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01418
UR  - https://arxiv.org/abs/2611.01418
AB  - Let K be a perfectoid field of characteristic p with countable value group (for example, K=β„‚_p^β™­). We study Zariski-closed subsets ZβŠ† 𝔹_K^n,perfdof the n-dimensional perfectoid closed unit ball 𝔹_K^n,perfd over K. We use an example of Gleason, and its generalization obtained in the authors' previous work, to show that two natural notions of dimension for such Zariski-closed subsets, the Krull dimension dim(Z) and Huber's iterated transcendence degree dim.tr(Z), do not coincide, and that the gap between them can be arbitrarily large.
ER  -