Bootstrapping to cyclotomic spectra

Last time we saw how to identify THH(ell^hZ)\otimes V with THH(ell^BZ)\otimes V at the level of spectra with Frobenius, for V a type 2 complex. I will explain how this spectrum level statement can be upgraded to the level of cyclotomic spectra, after replacing V with an arbitrary type 3 complex, and the Z-action by p^kZ for large k. I will then explain the Bockstein argument that shows that for a large enough k, the coassembly map for the T(2)-homology of TC(ell^hp^kZ) behaves as if the action were trivial. This finishes the proof that the T(2)-local TC of ell^{hp^kZ} is not K(2)-local.