Nonarchimedean bivariant $K$-theory

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, matrically stable, and excisive functors, similar to bivariant $K$-theory for locally convex topological $\mathbb{C}$-algebras and algebraic bivariant $K$-theory. As in the archimedean case, we use the universal property to construct a bivariant Chern character into analytic and periodic cyclic homology. When the first variable is the ground algebra $V$, we get a version of Weibel’s homotopy algebraic $K$-theory, which we call stabilised overconvergent analytic $K$-theory. The resulting analytic $K$-theory satisfies dagger homotopy invariance, stability by completed matrix algebras, and excision.