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Graded perfectoid rings

Ryo Ishizuka, Shou Yoshikawa

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 André's flatness lemma. These results lay the foundations for a graded theory of perfectoid rings.

Graded perfectoid rings

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 André's flatness lemma. These results lay the foundations for a graded theory of perfectoid rings.

Paper Structure

This paper contains 15 sections, 46 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and terminologies
  4. Perfectoid rings
  5. Graded homomorphisms
  6. Pro-graded rings
  7. Graded topological rings
  8. Pro-graded rings
  9. Graded perfectoid rings
  10. Graded perfectoid rings
  11. Graded André's flatness lemma
  12. Graded test perfectoid and its application
  13. Graded perfect prism and categorical equivalence
  14. Graded perfect prisms
  15. The proof of categorical equivalence

Key Result

Theorem A

Let $R$ be a $G$-graded ring. Then the following are equivalent:

Theorems & Definitions (108)

  • Definition 1.1: \ref{['DefGradedPerfectoid']} and \ref{['graded-perfect-prism']}
  • Theorem A: \ref{['except-for-p-comp', 'CatEquivPerfdPrism']}
  • Theorem B: \ref{['cat-equiv']}
  • Theorem C: \ref{['gr-perfdion']}
  • Theorem D: \ref{['Andre-graded']}
  • Theorem E: \ref{['GradedPerfdPure']} and \ref{['gr-p-pure']}
  • Definition 2.1: takaya2025Relative
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 98 more