245 research outputs found
Fr\'echet Modules and Descent
We study several aspects of the study of Ind-Banach modules over Banach rings
thereby synthesizing some aspects of homological algebra and functional
analysis. This includes a study of nuclear modules and of modules which are
flat with respect to the projective tensor product. We also study metrizable
and Fr\'{e}chet Ind-Banach modules. We give explicit descriptions of projective
limits of Banach rings as ind-objects. We study exactness properties of
projective tensor product with respect to kernels and countable products. As
applications, we describe a theory of quasi-coherent modules in Banach
algebraic geometry. We prove descent theorems for quasi-coherent modules in
various analytic and arithmetic contexts.Comment: improved versio
Dagger Geometry As Banach Algebraic Geometry
In this article, we apply the approach of relative algebraic geometry towards
analytic geometry to the category of bornological and Ind-Banach spaces
(non-Archimedean or not). We are able to recast the theory of Grosse-Kl\"onne
dagger affinoid domains with their weak G-topology in this new language. We
prove an abstract recognition principle for the generators of their standard
topology (the morphisms appearing in the covers). We end with a sketch of an
emerging theory of dagger affinoid spaces over the integers, or any Banach
ring, where we can see the Archimedean and non-Archimedean worlds coming
together
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