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

253where (P, ρ, g) is the pullback of ( l, ρ M) . Recall that P is the submodule ofdefined by settingC (M) × MP = { (x, m) ∈ C (M) × M | l (x) = ρ M (m) }and ρ : P → C (M) is defined by setting ρ ((x, m)) = x while g : P → M bysetting g ((x, m)) = m. Since ρ M is mono, ρ is also mono (thus ρ (P ) = H is asubcomodule of C (M) ) and since l is epi, g is epi. Denote by h M F : C(F ) → C (M) thecanonical inclusion for any F ⊆ M. Let m ∈ M, then there exist n ∈ N and F m ={y 1 , y 2 , . . . , y n } ⊆ M such that ρ M (m) = ∑ ))y∈F my ⊗ A c y = l(h M F m((c y ) y∈Fm.Then, for every m ∈ M, there exists z ∈ P such that m = g (z) = g ((ρ (z) , m))where ρ (z) = h M F m((c y ) y∈Fm)∈ h M F m(C(F m) ) ⊆ C (M) . Thus, for every m ∈ M, thereexists x m = (c y ) y∈Fm∈ such that m = g (( h M F m(x m ) , m )) . Then we have definedthe following homomorphismsuch thatν xm : x m A −→ Mx m ↦→ g (( h M F m(x m ) , m )) .m = ν xm (x m ) .Since x m A ⊆ C (Fm) , we deduce that the subcomodules of C (N) form a set of generatorsfor (Mod-A) C ⊕i.e. H is a generator for (Mod-A) C .H⊆C (N)(c) ⇒ (b) Obvious.(b) ⇒ (a) By Example 4.3 and Definition 4.10 F : Mod-A → (Mod-A) C is a rightadjoint of U and then F is left exact. Then using the hypothesis that U is left exact,we deduce that U ◦ F : Mod-A → Mod-A is also left exact, i.e. A C is flat. □Definition A.13. Let A be a Grothendieck category. An object A ∈ A is calledfinitely generated if, for every direct family of subobjects {A i } i∈Iof A such thatA = ∑ A i , there exists an index i 0 ∈ I such that A = A i0 .i∈IProposition A.14. Let A be a Grothendieck category. An object A ∈ A is finitelygenerated if and only if, for every family of subobjects {A i } i∈Iof A such that ∑ i∈IA i =∑A, there exists a finite number of subobjects A 1 , . . . , A n such that A = n A i .Proof. (⇐) Let {A i } i∈Ibe a direct family of subobjects of A closed under sumsand such that A = ∑ A i . By hypothesis there exists a finite number of subobjectsi∈I∑A 1 , . . . , A n such that A = n A i . Since the family is direct and closed under sums,i∈I∑there exists an index i 0 ∈ I such that A = n A i ⊆ A i0 . Then A is finitely generated.i∈Ii∈I