Grigory Andreychev | Searcharxiv

arXiv2023

$K$-Theorie adischer Räume

The present work is the author's doctoral thesis, written during his studies at the University of Bonn. Its goal is to establish the foundations of $K$-theory in the context of adic geometry using the formalism of condensed mathematics and Efimov's results on localizing invariants of dualizable categories. In particular, we study the descent properties of our version of $K$-theory and prove that it satisfies Nisnevich descent on analytic adic spaces. Furthermore, we show that $K$-theory forms a sheaf with respect to the étale topology after chromatic localization. Finally, we use this result to prove an analog of the Grothendieck-Riemann-Roch theorem for analytic adic spaces.

math.KT↗

arXiv2021

Pseudocoherent and Perfect Complexes and Vector Bundles on Analytic Adic Spaces

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.

math.AG↗