Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

264hence there exists a unique morphism g : P (X) → P (M) in A such that f =Hom A (P, −) (g). Let us consider in AP (X)g−→ P (M) → X → 0where X = Coker (g). Since P is projective, Hom A (P, −) is exact, and applying itwe get the exact sequenceHom A(P, P(X) ) f=Hom A (P,g)−→ Hom A(P, P(M) ) −→ Hom A (P, X) → 0.From this sequence and (276) we deduce that Q ≃ Hom A (P, X) where X =Coker (g) ∈ A.Conversely, let us assume that F : A → Mod-B is an equivalence of categories.Let G : Mod-B → A be its inverse equivalence. Since B is a progenerator and G isan equivalence of categories, by Proposition A.19 1) and 6), we deduce that G (B)is a progenerator in A. Moreover we haveB ≃ Hom Mod-B (B, B) ≃ Hom A (G (B) , G (B)) .Observe that G is a left adjoint to F and thus we haveHom A (G (B) , −) ≃ Hom Mod-B (B, F−) .Since Hom Mod-B (B, F−) ≃ F as functors, we deduce thatF ≃ Hom A (G (B) , −)where G (B) is a progenerator in A. Moreover, since G is an equivalence, G ( B (X)) ≃G (B) (X) .□