Pseudocoherent and Perfect Complexes and Vector Bundles on Analytic Adic Spaces

Abstract: Using the new approach to analytic geometry developed by Clausen and Scholze by means of condensed mathematics, we prove that for every affinoid analytic adic space $X$, pseudocoherent complexes, perfect complexes, and finite projective modules over $\mathcal{O}_X(X)$ form a stack with respect to the analytic topology on $X$; in particular, we prove that the category of vector bundles on $X$ is equivalent to the category of finite projective modules over $\mathcal{O}_X(X)$. To that end, we construct a fully faithful functor from the category of complete Huber pairs to the category of analytic rings in the sense of Clausen and Scholze and study its basic properties. Specifically, we give an explicit description of the functor of measures of the analytic ring associated to a complete Huber pair. We also introduce, following the ideas of Kedlaya-Liu, notions of \textit{pseudocoherent module} and \textit{pseudocoherent sheaf} in the context of adic spaces (where the latter can be regarded as an analogue of the notion of coherent sheaf in algebraic geometry) and show that there is a similar equivalence of the corresponding categories.