2025/08/15 by Dine, Dimitri
#14G22 #14G45 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Functional Analysis (math.FA) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2508.11268
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≤1 such that \varpi admits a compatible system of p-power roots \varpi^1/pn with ‖\varpi^1/pn‖=‖\varpi‖^1/pnfor all n, we prove that the "almost closed unit ball" functor M↦ M≤1ais 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≤1, (\varpi^1/p∞))-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≤1-module M≤1a, 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≤1, (\varpi^1/p∞))-modules in the sense of Mann, which factors through the full subcategory of solid condensed (A≤1, (\varpi^1/p∞))-almost modules. If A is perfectoid and the adic spectrum of (A, A∘) 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.