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PreprintarXiv licencearXiv

Decomposition Theorem for Perfectoid Rings along General Ideals

  • Ryo Ishizuka
  • Léo Navarro Chafloque
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Abstract

Using André's lemma and the excision square for perfectoidization coming from p-complete arc descent, we prove new structural results about perfectoid rings and perfectoidization. The main result is a tameness theorem for torsion in perfectoid rings: if R is a perfectoid ring and I⊂ R is an ideal, then the I-torsion in R is Iperfd-almost zero. This yields an excision-type decomposition of R along its I-torsion part. We also study (semi)perfectoid rings and perfectoid ideals and take the opportunity to make some structural remarks about them.

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Cite

BibTeXRIS
BibTeX
@misc{ishizuka2026decomposition,
  title = {{Decomposition Theorem for Perfectoid Rings along General Ideals}},
  author = {Ryo Ishizuka and Léo Navarro Chafloque},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.06241},
  eprint = {2606.06241},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.06241},
}
RIS
TY  - UNPB
TI  - Decomposition Theorem for Perfectoid Rings along General Ideals
AU  - Ryo Ishizuka
AU  - Léo Navarro Chafloque
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2606.06241
UR  - https://arxiv.org/abs/2606.06241
AB  - Using André's lemma and the excision square for perfectoidization coming from p-complete arc descent, we prove new structural results about perfectoid rings and perfectoidization. The main result is a tameness theorem for torsion in perfectoid rings: if R is a perfectoid ring and I⊂ R is an ideal, then the I-torsion in R is I_perfd-almost zero. This yields an excision-type decomposition of R along its I-torsion part. We also study (semi)perfectoid rings and perfectoid ideals and take the opportunity to make some structural remarks about them.
ER  -