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

258Proposition A.19. Let (T, H) be an adjunction where T : A → B, H : B → Aand A, B are Grothendieck categories. If T is an equivalence of categories, then1) if P is a generator of A then T P is a generator of B2) if P is projective in A then T P is projective B3) if P is finite in A then T P is finite B4) if P is finitely generated in A then T P is finitely generated in B5) if P is finitely generated and projective in A then T P is finitely generatedand projective B6) if P is finite projective in A then T P is finite projective in B.Proof. 1) Let f : Y → Y ′ be a non zero morphism in B. Since T is an equivalence,there exists a non zero morphism g : X → X ′ in A such that f = T (g). Since Pis a generator of A there exists a morphism p : P → X such that g ◦ p ≠ 0. Then0 ≠ T (g ◦ p) = T (g) ◦ T (p) = f ◦ T (p) and T (p) : T P → T X = Y so that T P is agenerator of B.2) Let f : Y → Y ′ be a morphism in B. Since T is an equivalence, there existsa morphism g : X → X ′ in A such that f = T (g), Y = T X and Y ′ = T X ′ . Letl : T P → Y a morphism in B, then l : T P → T X then there exists h : P → X suchthat l = T (h). Since P is projective A, there exists k : P → X ′ such that g ◦ h = k.By applying the functor T we get T (g ◦ h) = T (g) ◦ T (h) = f ◦ l = T (k) then T Pis projective in B.3) Let (N i ) i∈Ibe a family of objects in B. Since T is an equivalence, there existsa family (M i ) i∈Iof objects in A such that (N i ) i∈I= (T M i ) i∈I. Denote by ε i :M i → ∐ (M i and by Hom A (P, ε i ) : Hom A (P, M i ) → Hom A P, ∐ M i). Then wei∈Ii∈I (can consider the codiagonal morphism ∇ (Hom A (P, ε i )) i∈I: Hom A P, ∐ )M i →∐i∈IHom A (P, M i ). Since P is finite in A we have that P preserves coproducts, i.e.i∈I∇ (Hom A (P, ε i )) i∈Iis an isomorphism. Letφ X,X ′ : Hom A (X, X ′ ) −→ Hom B (T X, T X ′ )f ↦→ T (f)Since T is an equivalence φ X,X ′ is bijective for every X, X ′ ∈ A and ∐ (T M i ) =( )i∈I∐T M i . Then we can consideri∈I ( )) (∐Hom B (T P, T (ε i )) : Hom B (T P, T (M i )) → Hom B(T P, T M i = Hom B T P, ∐i∈I (i∈Iand their codiagonal morphism ∇ (Hom B (T P, T (ε i ))) i∈I: Hom B T P, ∐ )(T M i ) →i∈I)(T M i )