  We show under mild hypotheses that a Quillen adjunction between stable model
categories induces another Quillen adjunction between their left localizations,
and we provide conditions under which the localized adjunction is a Quillen
equivalence. Moreover, we show that in many cases the induced Quillen
equivalence is symmetric monoidal. Using our results we construct a symmetric
monoidal algebraic model for rational cofree $G$-spectra. In the process, we
also show that $L$-complete modules provide an abelian model for derived
complete modules.
