A Cartesian diagram of Rapoport–Zink towers over universal covers of $p$-divisible groups

In “J. Moduli of $p$-divisible groups” [9], 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 (“E. Cycles in the cohomology of Rapoport–Zink towers coming from the Lubin–Tate tower” [6]) to compute certain non-trivial $p$-adic étale cohomology classes appearing in the the generic fiber of Lubin–Tate and Rapoport–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.

2010 Mathematics Subject Classification

11G18, 11G25, 11Sxx

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Received 31 January 2020

Accepted 13 March 2020

Published 2 October 2020