Fix an odd prime $p$. Let $X$ be a pointed space whose $p$-completed K-theory $\mathrm{KU}_p^*(X)$ 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 $\mathbf{E}_\infty$-$\mathrm{KU}_p$-algebra spectrum $\mathrm{KU}_p^{X_+}$ of $\mathrm{KU}_p$-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 $X$.
This is essentially a cleaned up version of my Master’s thesis, with a major mistake corrected. The correction lead to a joint appendix with Max Blans, where we describe an analogue of the Tor spectral sequence in the $\mathrm{K}(1)$-local setting.