Smash products of E(1)-local spectra at an odd prime

In 1996, Franke constructed a purely algebraic category that is equivalent as a triangulated category to the E(n)-local stable homotopy category for n^2+n < 2p-2. The two categories are not Quillen equivalent, and his proof uses systems of triangulated diagram categories rather than model categories. Our main result is that in the case n=1 Franke's functor maps the derived tensor product to the smash product. It can however not be an associative equivalence of monoidal categories. The first part of our paper sets up a monoidal version of Franke's systems of triangulated diagram categories and explores its properties. The second part applies these results to the specific construction of Franke's functor in order to prove the above result.

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