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

246We haveT (ζ) ◦ ξ ◦ T H (p) (262)= T (ζ) ◦ T (q) (259)= T H (p)and since T H (p) is epi by Lemma A.5, we getWe computeand since T (q) is epi, we obtain thatT (ζ) ◦ ξ = Id T H(Coim(f)) .ξ ◦ T (ζ) ◦ T (q) (259)= ξ ◦ T H (p) (262)= T (q)ξ ◦ T (ζ) = Id T (Coim(H(f))) .Therefore T (ζ) is an isomorphism. From formula (261) we get that T H (i) ◦ T (ζ) =T (j) and by Lemma A.5 we conclude that T (j) is a monomorphism.Let m, n ∈ N, m, n ≥ 1, let f : B m → B n be a morphism of right B-modulesand let X = Coim ( f ) . Since U is a generator, by Proposition A.3, there existsa unique morphism f : U m → U n such that H (f) = f. Then, by the foregoing,X = Coim (H (f)) and T (j) is a monomorphism where j : X → B n is the canonicalmonomorphism.□Lemma A.7. Let B be a ring and let X be a submodule of a free module A (Z) .Let P 0 (X) be the set of finite subsets of X and let j F : X F → X be the canonicalinclusion of the submodule of X spanned by F ∈ P 0 (X). Then for every F ∈ P 0 (X)there exists a finite subset Z F of Z and a monomorphism i F : X F → A (Z F ) such thatthe diagramj FX F X i Fj h FA (Z F )A (Z)where j : X → A (Z) is the canonical inclusion and h F : A (Z F ) → A (Z) is the canonicalsection of the canonical projection π F : A (Z) → A (Z F ) , is commutative. Moreover(i F ) F ∈P0 (X) is a family of morphisms between the direct systems (X F )( ) F ∈P0 (X) andA(Z F ) and (h F ∈P 0 (X) F ◦ i F ) F ∈P0 (X)is a compatible family of morphisms such thatlim (h F ◦ i F ) = j.−→Proof. Let (e z ) z∈Zbe the canonical basis of A (Z) . Then, for every x ∈ X there existsa finite subset F x of Z such thatFor every F ∈ P 0 (X) let us setx = ∑ z∈F xe z a z where a z ∈ A for every z ∈ F x .Z F = ⋃ x∈FF xand let ( e F z)z∈Z Fbe the canonical basis of A (Z F ) . Then the assignmente F z↦→ e z where z ∈ Z F