The homotopy category of flat modules, and Grothendieck duality

Let R be a ring. We prove that the homotopy category K ( R -Proj) is always -compactly generated, and, depending on the ring R , it may or may not be compactly generated. We use this to give a description of K ( R -Proj) as a quotient of K ( R -Flat). The remarkable fact is that this new description of K ( R -Proj) generalizes to non-affine schemes; this will appear in Murfet’s thesis.