Decomposition Theorem for Perfectoid Rings along General Ideals

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.

26 pages. Added results in the perfectoid ideals section. Comments welcome