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arXiv · 2606.06241

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 $I\subset R$ is an ideal, then the $I$-torsion in $R$ is $I_{\mathrm{perfd}}$-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.

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Ryo Ishizuka, Léo Navarro Chafloque. 2026-07-03. Decomposition Theorem for Perfectoid Rings along General Ideals. https://arxiv.org/abs/2606.06241

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