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 $\aleph_1$ -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.