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As a step towards understanding the $\mathrm{tmf}$-based Adams spectralsequence, we compute the $K(1)$-local homotopy of $\mathrm{tmf} \wedge\mathrm{tmf}$, using a small presentation of $L_{K(1)}\mathrm{tmf}$ due toHopkins. We also describe the $K(1)$-local $\mathrm{tmf}$-based Adams spectralsequence.

On the $K(1)$-local homotopy of $\mathrm{tmf} \wedge \mathrm{tmf}$