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Perfectoid towers generated from prisms

Ryo Ishizuka

Abstract

We present a unified construction of perfectoid towers from specific prisms which covers all the previous constructions of (p-torsion-free) perfectoid towers. By virtue of the construction, perfectoid towers can be systematically constructed for a large class of rings with Frobenius lift. Especially, any Frobenius lifting of a reduced $\mathbb{F}_p$-algebra has a perfectoid tower.

Perfectoid towers generated from prisms

Abstract

We present a unified construction of perfectoid towers from specific prisms which covers all the previous constructions of (p-torsion-free) perfectoid towers. By virtue of the construction, perfectoid towers can be systematically constructed for a large class of rings with Frobenius lift. Especially, any Frobenius lifting of a reduced -algebra has a perfectoid tower.
Paper Structure (16 sections, 31 theorems, 69 equations)

This paper contains 16 sections, 31 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Prisms
  3. Construction of Towers
  4. Construction of Towers from Prisms
  5. Construction of Frobenius Projections
  6. Construction of Perfectoid Towers from Prisms
  7. Perfectoid Towers from delta-rings
  8. Base change by perfectoid towers
  9. Adjoining $p$-power roots of $p$ and unity
  10. Generic ranks of transition maps of perfectoid towers
  11. Examples
  12. Previous examples of perfectoid towers
  13. Examples from geometric Frobenius lifts
  14. Examples from affine semigroups
  15. Examples from $\delta$-stable ideals
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(A, (d))$ be an (orientable) prism such that $p, d$ is a regular sequence on $A$ and $A/pA$ is $p$-root closedA ring $A/pA$ is $p$-root closed in $A/pA[1/d]$ if $x \in A/pA[1/d]$ satisfies $x^{p^n} \in A/pA$ for some $n \geq 1$, then $x \in A/pA$ holds. in $A/pA[1/d]$. Then the tower of rings induced from the Frobenius lift $\varphi\colon A\to A$ becomes a perfectoid tower (PerfectoidTower)

Theorems & Definitions (83)

  • Theorem 1.1: Special case of \ref{['PerfectoidTowerPrism']}
  • Theorem 1.2: Special case of \ref{['ExplicitPerfectoidTowerDeltaRing']}
  • Example 1.3
  • Definition 2.1: bhatt2022Prismsa
  • Definition 2.2: bhatt2022Prismsa
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 3.2
  • ...and 73 more