Representability of continuous K-theory in rigid analytic motivic ${\mathbb{A}}^1$-homotopy theory

Abstract

We prove that both continuous K-theory and analytic K-theory of rigid analytic spaces (à la Kerz--Saito--Tamme) satisfiy descent with respect to the Nisnevich topology. Together with the fact that it is ${\mathbb{A}}^1$-invariant assuming resolutions of singularities, we deduce that it is representable in the ${\mathbb{A}}^1$-homotopy category of rigid spaces (à la Dahlhausen--Yaylali). We identifiy the representing object with both $\mathbb{Z}\times \mathrm{BGL}$ and the analytification of algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra. Moreover, we show Weibel vanishing and that continuous K-theory is ${\mathbb{A}}^1$-invariant on local Tate pairs (without any regularity assumption).

arXiv comment52 pages. Split-off from arXiv:2407.09606v2 with new coauthor, corrected proofs, and new results. Comments very welcome!