Banach modules, almost mathematics and condensed mathematics
Dimitri Dine

TL;DR
This paper explores the deep connections between almost mathematics, condensed mathematics, and Banach modules over Banach rings, establishing equivalences and embeddings that reveal how norms are determined by almost structures, especially in the perfectoid setting.
Contribution
It introduces a new equivalence between Banach modules and almost modules, showing the norm is determined by the almost structure, and constructs a fully faithful embedding into condensed almost modules.
Findings
The almost closed unit ball functor is an equivalence of categories.
The norm on a Banach module is determined by its almost module structure.
Embedding of Banach modules into condensed almost modules preserves tensor products in the perfectoid case.
Abstract
We study the relationship between almost mathematics, condensed mathematics and the categories of seminormed and Banach modules over a Banach ring , with submetric (norm-decreasing) -module homomorphisms for morphisms. If is a Banach ring with a norm-multiplicative topologically nilpotent unit contained in the closed unit ball such that admits a compatible system of -power roots with \begin{equation*}\lVert\varpi^{1/p^{n}}\rVert=\lVert\varpi\rVert^{1/p^{n}}\end{equation*}for all , we prove that the "almost closed unit ball" functor \begin{equation*}M\mapsto M_{\leq1}^{a}\end{equation*}is an equivalence between the category of Banach -modules and submetric -module maps and the category of -adically complete, -torsion-free almost -modules. We also obtain an…
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