Graded perfectoid rings

TL;DR

This paper develops a graded theory of perfectoid rings, establishing key equivalences and constructions that extend existing ungraded perfectoid theory to a graded setting, thus laying foundational groundwork.

Contribution

It introduces graded perfectoid rings, proves their equivalence with graded perfect prisms, and constructs initial graded perfectoid covers, extending foundational perfectoid theory.

Findings

Established categorical equivalence between graded perfectoid rings and graded perfect prisms.

Constructed initial graded perfectoid covers for graded semiperfectoid rings.

Proved a graded version of André's flatness lemma.

Abstract

We introduce and study graded perfectoid rings as graded analogues of Scholze's (integral) perfectoid rings. We establish a categorical equivalence between graded perfectoid rings and graded perfect prisms, extending the Bhatt-Scholze's correspondence to the graded setting. We also construct the initial graded perfectoid cover of any graded semiperfectoid rings and prove a graded version of Andr\'e's flatness lemma. These results lay the foundations for a graded theory of perfectoid rings.