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

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.