Fréchet modules and descent

Motivated by classical functional analysis results over the complex numbers and results in the bornological setting over the complex numbers of R. Meyer, we study several aspects of the study of Ind-Banach modules over Banach rings. This allows for a synthesis of 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 Fréchet Ind-Banach modules. We give explicit descriptions of projective limits of Banach rings as ind-objects. We study exactness properties of the 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 and relate them to well known complexes of modules coming from covers.