Explicit Cogenerators for the Homotopy Category of Projective Modules over a Ring

Abstract

Let R be a ring. In two previous articles [12, 14] we studied the homotopy category K(R-Proj) of projective R-modules. We produced a set of generators for this category, proved that the category is N1-compactly generated for any ring R, and showed that it need not always be compactly generated, but is for sufficiently nice R We furthermore analyzed the inclusion j!: K(R-Proj) → K(R-Flat) and the orthogonal subcategory & = K(R-Proj). And we even showed that the inclusion & → K(R-Flat) has a right adjoint; this forces some natural map to be an equivalence K(R-Proj) → &. In this article we produce a set of cogenerators for K(R-Proj). More accurately, this set of cogenerators naturally lies in the equivalent & ≅ K(R-Proj); it can be used to give yet another proof of the fact that the inclusion & → K(R-Flat) has a right adjoint. But by now several proofs of this fact already exist.