The K-theory cochains of H-spaces and height 1 chromatic homotopy theory

Fix an odd prime $p$. Let $X$ be a pointed space whose $p$-completed K-theory ${\mathrm{K}\mathrm{U}}_p^{\ast }(X)$ is an exterior algebra on a finite number of odd generators; examples include odd spheres and many H-spaces. We give a generators-and-relations description of the ${\mathbf{E}}_{\mathrm{\infty }}$-${\mathrm{K}\mathrm{U}}_p$-algebra spectrum ${\mathrm{K}\mathrm{U}}_p^{X_{+}}$ of ${\mathrm{K}\mathrm{U}}_p$-cochains of $X$. To facilitate this construction, we describe a $\mathrm{K}(1)$-local analogue of the Tor spectral sequence for ${\mathbf{E}}_1$-ring spectra. Combined with previous work of Bousfield, this description of the cochains of $X$ recovers a result of Kjaer that the $v_1$-periodic homotopy type of $X$ can be modelled by these cochains. This then implies that the Goodwillie tower of the height 1 Bousfield-Kuhn functor converges for such $X$.