I'm terribly confused about finite and smashing localizations. I thought being a finite localization is equivalent to the inclusion of acyclics preserving compact objects but I just now realized that I can "prove" with this definition that smashing implies finite, like this:
Let $L$ be a smashing localization functor and let $G$ be the corresponding acyclization (both as endofunctors of spectra). By the equivalent characterization of smashing $G$ preserves colimits and therefore the right adjoint to the inclusion of the acyclics must preserve colimits as well. This is equivalent to the inc…