Abstract: In a series of three papers, in 2005 and 2006, Jorgensen, Krause and Iyengar noticed that dualizing complexes, in the sense of Grothendieck, are related to certain functors between homotopy categories. The most intriguing connection is the one with the homotopy category of flat modules. We will begin by explaining the recent work of Jorgensen, Krause and Iyengar, and then some of my results which followed from their work. All of these papers work only on affine schemes; they are statements about modules over a ring R.

Very recently Daniel Murfet was able to extend much of the theory to the global case, where we may have a scheme which is not affine. We will try to survey his work.