Franke's Realization Functor and Monoidal Products

In 1996, Jens Franke in an unpublished paper states that the homotopy category of E(1)-local spectra is equivalent as a triangulated category to D1(A), the derived category of quasi-periodic cochain complexes of period 1 for primes p ≥ 3. This is Franke's realization functor R: D1(A) → Ho(L1Sp). However, Irakli Patchkoria spotted gaps in the proof of J.Franke that were filled in a series of papers and put in a firm ground that for primes p ≥ 5 Franke's realization functor is a triangulated equivalence. The categories D1(A) and v are in fact tensor-triangulated, that is, both categories posses a monoidal structure that are compatible with the triangulated structure. In this thesis we prove that Franke's realization functor commutes with the monoidal products up to a natural isomorphism, that is, R i is tensor triangulated functor.