The very effective covers of KO and KGL over Dedekind schemes

Abstract

We answer a question of Hoyois–Jelisiejew–Nardin–Yakerson regarding framed models of motivic connective $K$-theory spectra over Dedekind schemes. That is, we show that the framed suspension spectrum of the presheaf of groupoids of vector bundles (resp. non-degenerate symmetric bilinear bundles) is the effective cover of KGL (resp. very effective cover of KO). One consequence is that, over any scheme, we obtain a spectral sequence from Spitzweck’s motivic cohomology to homotopy algebraic $K$-theory; it is strongly convergent under mild assumptions.