Cohomology theories based on flats

Let A be an associative ring with identity, the homotopy category of flat modules and the full subcategory of pure complexes. The quotient category , called here the pure derived category of flats, was introduced by Neeman. In this category flat resolutions are unique up to homotopy and so can be used to compute cohomology. We develop theories of Tate and complete cohomology in the pure derived category of flats. These theories extend naturally to sheaves over semi-separated noetherian schemes, where there are not always enough projectives, but we do have enough flats. As applications we characterize rings with finite sfli and schemes which are locally Gorenstein.