Neeman's characterization of K(R-Proj) via Bousfield localization

Let $R$ be an associative ring with unit and denote by $K({\rm R \mbox{-}Proj})$thehomotopycategoryofcomplexesofprojectiveleft$R$-modules\mathrm{.}Neemanprovedthetheoremthat$K({\rm R \mbox{-}Proj})$is$\aleph_1$-compactlygenerated,withthecategory$K^+ ({\rm R \mbox{-}proj})$ofleftboundedcomplexesoffinitelygeneratedprojective$R$-modules providing an essentially small class of such generators. Another proof of Neeman’s theorem is explained, using recent ideas of Christensen and Holm, and Emmanouil. The strategy of the proof is to show that every complex in $K({\rm R \mbox{-}Proj})$vanishesintheBousfieldlocalization$K({\rm R \mbox{-}Flat})/\langle K^+ ({\rm R \mbox{-}proj}) \rangle.$