Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Perfectoid Spaces in Multivariate $p$-adic Hodge Theory

Aprameyo Pal, Rohit Pokhrel

Abstract

Perfectoid spaces have become a crucial tool in $p$-adic geometry, serving as a bridge between adic spaces in characteristic $0$ and those in characteristic $p$. In this article, we develop a systematic way to study the structure of perfectoid spaces within the setting of multivariate $p$-adic Hodge theory over a variant of the rings introduced in \cite{Bri}.

Perfectoid Spaces in Multivariate $p$-adic Hodge Theory

Abstract

Perfectoid spaces have become a crucial tool in -adic geometry, serving as a bridge between adic spaces in characteristic and those in characteristic . In this article, we develop a systematic way to study the structure of perfectoid spaces within the setting of multivariate -adic Hodge theory over a variant of the rings introduced in \cite{Bri}.

Paper Structure

This paper contains 12 sections, 47 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Notations
  4. Main Results
  5. Acknowledgment
  6. Objects in mixed characteristics setups
  7. Objects in equicharacteristics setups
  8. Non-Archimedean functional analysis over $K_\Delta$
  9. Almost Mathematics in multivariate setups
  10. Perfectoid Algebras over $K_\Delta$
  11. Perfectoid Spaces over $K_\Delta$
  12. Almost Purity Theorem for Perfectoid Spaces over $K_\Delta$

Key Result

Theorem 1.2

The categories $(K_\Delta)_{\mathop{\mathrm{f\acute{e}t}}\nolimits}$ and $(K^{\flat}_{\Delta})_{\mathop{\mathrm{f\acute{e}t}}\nolimits}$ are equivalent. This runs through the following chain of equivalence $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (121)

  • Definition 1.1: Sch, Definition 1.2
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 111 more