Banach modules, almost mathematics and condensed mathematics

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 $A$, with submetric (norm-decreasing) $A$-module homomorphisms for morphisms. If $A$ is a Banach ring with a norm-multiplicative topologically nilpotent unit $\varpi$ contained in the closed unit ball $A_{\leq1}$ such that $\varpi$ admits a compatible system of $p$-power roots $\varpi^{1/p^{n}}$ with \begin{equation*}\lVert\varpi^{1/p^{n}}\rVert=\lVert\varpi\rVert^{1/p^{n}}\end{equation*}for all $n$, 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 $A$-modules and submetric $A$-module maps and the category of $\varpi$-adically complete, $\varpi$-torsion-free almost $(A_{\leq1}, (\varpi^{1/p^{\infty}}))$-modules. We also obtain an analogous result for Banach algebras and almost algebras. The main novelty in our approach is that we show that the norm on the Banach module $M$ is completely determined by the corresponding almost $A_{\leq1}$-module $M_{\leq1}^{a}$, rather than being determined only up to equivalence. We deduce from our results the existence of a natural fully faithful embedding of the category of Banach $A$-modules and submetric $A$-module maps into the category of (static) condensed almost $(A_{\leq1}, (\varpi^{1/p^{\infty}}))$-modules in the sense of Mann, which factors through the full subcategory of solid condensed $(A_{\leq1}, (\varpi^{1/p^{\infty}}))$-almost modules. If $A$ is perfectoid and the adic spectrum of $(A, A^{\circ})$ is totally disconnected, we show that this embedding transforms the complete tensor product of Banach $A$-modules into (an almost analog of) the solid tensor product of solid condensed almost modules.