kq-resolutions I

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 Röndigs-Spitzweck-Østvær 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.