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

254(⇒) Let {A i } i∈Ibe a family of subobjects of A closed under sums and whichcontains A itself. Assume that A = ∑ A i . Since A is finitely generated, there existsi∈Ian index i 0 ∈ I such that A i0 = A.Lemma A.15. Let A be a Grothendieck category and let 0 → A ′ −→ A −→ pA ′′ →0 be an exact sequence in A. Then if A is finitely generated A ′′ is also finitelygenerated.Proof. Let (A ′′i ) i∈Ibe a direct family of subobjects of A ′′ such that A ′′ = ∑ A ′′i . Theni∈Iwe have, for every i ∈ I, A ′′i = A i A ′ for A i subobject of A such that A ′ ⊆ A i .Hence (A i ) i∈Iis a direct family of subobjects of A such that A = ∑ A i and since Ai∈Iis finitely generated there exists an index i 0 ∈ I such that A = A i0 . Then we haveA ′′ = A i0 A ′ = A ′′i 0, i.e. A ′′ is also finitely generated.□Lemma A.16. Let A be a Grothendieck category and let A ∈ A be a finitely generatedobject. Let f : A → ∐ i∈IB i be a morphism in A. Then there exist a finite subsetF ⊆ I such that Im (f) ⊆ ∑ ε i (B i ).i∈F)B ii∈Ii∈IProof. Let ε i : B i → ∐ B j the canonical inclusions and consider ∇ (ε i ) i∈I: ∐ B i →j∈Ii∈I( )∐∐B j defined by setting ∇ (ε i ) i∈IB j = ∑ ε i (B i ). We prove ∇ (ε i ) i∈I=j∈I j∈Ii∈IId`B i. In fact we have that ∇ (ε j ) j∈I◦ ε i = ε i = Id ` B j◦ ε i . Thus, ∐ B i =i∈Ij∈I(i∈IIm Id` = Im ( ) ∑∇ (ε i ) i∈I = ε i (B i ) where (ε i (B i )) i∈Idefine a family of subobjectsof ∐ B i . Let f : A → ∐ B j . By Lemma A.15, since A is finitely generatedi∈Ij∈Ialso Coim (f) is finitely generated and, since Coim (f) ≃ Im (f) ⊆ ∑ ε i (B i ), therei∈I□exists a finite subset F ⊆ I such that Im (f) ⊆ ∑ ε i (B i ).i∈F□Definition A.17. Let A be an abelian category. A projective object P ∈ A iscalled finite if the functor Hom A (P, −) preserves coproducts.Proposition A.18. Let A be a Grothendieck category and let P be a projectiveobject. Then P is finite if and only if P is finitely generated.Proof. Assume first that P ∈ A is finite. Let {P i } i∈Ibe a family of subobjects of Psuch that ∑ i∈IP i = P. Let p i : P i → P be the canonical inclusion for every i ∈ I andconsider p = ∇ (p i ) i∈I: ∐ i∈IP i → P. Then we have( ) ( )∐∇ (pi ) i∈IP i = ∑i∈Ii∈Ip i (P i ) = ∑ i∈IP i = P