Algebraization of absolute perfectoidization via section rings

TL;DR

The paper develops a graded approach to absolute perfectoidization for G-graded adic rings and demonstrates algebraization of the process for structure sheaves of certain formal schemes.

Contribution

It introduces a graded framework for absolute perfectoidization and proves algebraization results for structure sheaves in a geometric setting.

Findings

Constructed a graded version of absolute perfectoidization for G-graded adic rings.

Proved algebraization of the absolute perfectoidization of structure sheaves of projective-type formal schemes.

Abstract

We construct and study a graded version of absolute perfectoidization for $G$-graded adic rings. As a main geometric application, we show that the absolute perfectoidization of the structure sheaf of a projective-type formal scheme admits an algebraization.