Nonarchimedean bivariant K-theory

TL;DR

This paper develops a new bivariant K-theory framework for nonarchimedean bornological algebras over a complete discrete valuation ring, extending concepts from algebraic and topological K-theory to the nonarchimedean setting.

Contribution

It introduces a universal bivariant K-theory for nonarchimedean bornological algebras, including a version of homotopy algebraic K-theory called stabilised overconvergent analytic K-theory.

Findings

The new theory is dagger homotopy invariant.

It is stable under completed matrix algebra operations.

It satisfies excision property.

Abstract

We introduce bivariant K-theory for nonarchimedean bornological algebras over a complete discrete valuation ring $V$. This is the universal target for dagger homotopy invariant, matricially stable and excisive functors, similar to bivariant K-theory for locally convex topological $\mathbb{C}$-algebras and algebraic bivariant K-theory. When the first variable is the ground algebra $V$, we get a version of Weibel's homotopy algebraic K-theory, which we call \textit{stabilised overconvergent analytic K-theory}. The resulting analytic K-theory satisfies dagger homotopy invariance, stability by completed matrix algebras, and excision.