K-theory of non-archimedean rings I

TL;DR

This paper introduces analytic K-theory for Tate rings, establishing its homotopy invariance, proving an analog of the Bass fundamental theorem, and comparing it with continuous K-theory, thus advancing the understanding of K-theory in non-archimedean settings.

Contribution

It develops a new variant of homotopy K-theory called analytic K-theory for Tate rings, with key theorems and comparisons to existing theories.

Findings

Analytic K-theory is homotopy invariant for Tate rings.

Proved an analytic analog of the Bass fundamental theorem.

Compared analytic K-theory with continuous K-theory.

Abstract

We introduce a variant of homotopy K-theory for Tate rings, which we call analytic K-theory. It is homotopy invariant with respect to the analytic affine line viewed as an ind-object of closed disks of increasing radii. Under a certain regularity assumption we prove an analytic analog of the Bass fundamental theorem and we compare analytic K-theory with continuous K-theory, which is defined in terms models. Along the way we also prove some results about the algebraic K-theory of Tate rings.