$kq$-Resolutions I

Abstract

Let $kq$ denote the very effective cover of Hermitian K-theory. We apply the $kq$-based motivic Adams spectral sequence, or $kq$-resolution, to computational motivic stable homotopy theory. Over base fields of characteristic not two, we prove that the $n$-th stable homotopy group of motivic spheres is detected in the first $n$ lines of the $kq$-resolution, thereby reinterpreting results of Morel and R{\"o}ndigs-Spitzweck-{\O}stv{\ae}r in terms of $kq$ and $kq$-cooperations. Over algebraically closed fields of characteristic 0, we compute the ring of $kq$-cooperations modulo $v_1$-torsion, establish a vanishing line of slope $1/5$ in the $E_2$-page, and completely determine the $0$- and $1$- lines of the $kq$-resolution. This gives a full computation of the $v_1$-periodic motivic stable stems and recovers Andrews and Miller's calculation of the $\eta$-periodic $\mathbb{C}$-motivic stable stems. We also construct a motivic connective $j$ spectrum and identify its homotopy groups with the $v_1$-periodic motivic stable stems. Finally, we propose motivic analogs of Ravenel's Telescope and Smashing Conjectures and present evidence for both.