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

TL;DR

This paper generalizes a Cartesian diagram of perfectoid spaces related to Rapoport-Zink towers, enabling new computations in p-adic Hodge theory and étale cohomology, with implications for the structure of vector bundles on the Fargues-Fontaine curve.

Contribution

It extends the Cartesian diagram result of Scholze and Weinstein to broader settings, facilitating advanced cohomological calculations in p-adic geometry.

Findings

Generalized the Cartesian diagram to new contexts

Enabled computation of non-trivial étale cohomology classes

Analyzed the behavior of the vector bundle functor under multilinear morphisms

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 \'etale 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.