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Perfectoid rings as Thom spectra

Zhouhang Mao

Abstract

The Hopkins-Mahowald theorem realizes the Eilenberg-Maclane spectra $H\mathbb F_p$ as Thom spectra for all primes $p\in\mathbb N_{>0}$. In this article, we record a known proof of a generalization of Hopkins-Mahowald theorem, realizing $Hk$ as Thom spectra for perfect rings $k$, and we provide a further generalization by realizing $HR$ as Thom spectra for perfectoid rings $R$. We also discuss even further generalizations to prisms $(A,I)$ and indicates how to adapt our proofs to Breuil-Kisin case.

Perfectoid rings as Thom spectra

Abstract

The Hopkins-Mahowald theorem realizes the Eilenberg-Maclane spectra as Thom spectra for all primes . In this article, we record a known proof of a generalization of Hopkins-Mahowald theorem, realizing as Thom spectra for perfect rings , and we provide a further generalization by realizing as Thom spectra for perfectoid rings . We also discuss even further generalizations to prisms and indicates how to adapt our proofs to Breuil-Kisin case.

Paper Structure

This paper contains 20 sections, 88 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Recollection of spherical Witt vectors
  3. Perfect rings being Thom spectra
  4. Recollection of perfectoid rings
  5. Basic definitions and properties
  6. Universal properties of Fontaine's map (and a spherical analogue)
  7. Proof of the main theorem
  8. Finiteness and completeness of $M f_{R, \xi}$ and $H R$ as $W^+ (R^{\flat})$-modules
  9. $t_{R, \xi}$ is an equivalence after the base change along $W^+ (R^{\flat}) \rightarrow W^+ (\kappa)$
  10. $t_{R, \xi}$ is an equivalence after the base change along $W^+ (R^{\flat}) \rightarrow H R^{\flat}$
  11. Conclude: $t_{R, \xi}$ is an equivalence
  12. An intermezzo: Identifying $\operatorname{THH} \left( - / W^+ (k) \right)$ with $\operatorname{THH} \left( - \right)$ after $p$-completion
  13. Analogues
  14. Preparations
  15. The Breuil-Kisin case
  16. ...and 5 more sections

Key Result

Proposition 1.3

$\pi_1 (\operatorname{BGL}_1 (R)) = \operatorname{GL}_1 (\pi_0 R)$ for any ring spectra $R$. Concretely, an invertible element $a \in \pi_0 R$ corresponds to a multiplication map $m_a : R \rightarrow R$ in $\operatorname{BGL}_1 (R)$.

Figures (1)

  • Figure 1:

Theorems & Definitions (181)

  • Definition 1.1
  • Remark 1.2
  • Proposition 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 171 more