# $K$-Theory of Non-Archimedean Rings. I

### Moritz Kerz

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany### Shuji Saito

Graduate School of Mathematical, Sciences University of Tokyo, 3-8-1 Komaba, Tokyo, Japan### Georg Tamme

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

## 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 of models. Along the way we also prove some results about the algebraic $K$-theory of Tate rings.

## Cite this article

Moritz Kerz, Shuji Saito, Georg Tamme, $K$-Theory of Non-Archimedean Rings. I. Doc. Math. 24 (2019), pp. 1365–1411

DOI 10.4171/DM/707