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A Cartesian Diagram of Rapoport-Zink Towers over Universal Covers of $p$-Divisible Groups

Mohammad Hadi Hedayatzadeh

Abstract

In their paper Scholze and Weinstein show that a certain diagram of perfectoid spaces is Cartesian. In this paper, we generalize their result. This generalization will be used in a forthcoming paper of ours to compute certain non-trivial $\ell$-adic étale cohomology classes appearing in the the generic fiber of Lubin-Tate and Rapoprt-Zink towers. We also study the behavior of the vector bundle functor on the fundamental curve in $p$-adic Hodge theory, defined by Fargues-Fontaine, under multilinear morphisms.

A Cartesian Diagram of Rapoport-Zink Towers over Universal Covers of $p$-Divisible Groups

Abstract

In their paper Scholze and Weinstein show that a certain diagram of perfectoid spaces is Cartesian. In this paper, we generalize their result. This generalization will be used in a forthcoming paper of ours to compute certain non-trivial -adic étale cohomology classes appearing in the the generic fiber of Lubin-Tate and Rapoprt-Zink towers. We also study the behavior of the vector bundle functor on the fundamental curve in -adic Hodge theory, defined by Fargues-Fontaine, under multilinear morphisms.

Paper Structure

This paper contains 11 sections, 39 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rank and the exterior power of a matrix
  4. Elements from $p$-adic Hodge theory
  5. Universal cover of $p$-divisible groups
  6. Multilinear Theory
  7. Multilinear morphisms of graded modules
  8. Multilinear morphisms of vector bundles on the Fargues-Fontaine curve
  9. Multilinear theory of $p$-divisible groups
  10. The wedge morphism on the Lubin-Tate tower
  11. Main Theorem

Key Result

Lemma 2.3

Let $\varphi:R\rightarrow S$ be a ring homomorphism, then for any $A\in{\mathbb{M}}_{n}(R)$, and any $i\geq 0$, we have

Theorems & Definitions (101)

  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • ...and 91 more