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

and thus p is an epimorphism. Since P is projective there exists i : P → ∐ P i such(i∈Ithat p ◦ i = Id P . Note that i ∈ Hom A P, ∐ )P i and since P is finite we havei∈Ithat ∇ (Hom (P, ε i )) i∈I: ∐ (Hom A (P, B i ) → Hom A P, ∐ )B i is an isomorphism.i∈Ii∈IThus there exist n ∈ N and i 1 ∈ Hom A (P, P 1 ) , . . . , i n ∈ Hom A (P, P n ) such that∐i = ε 1 i 1 + · · · + ε n i n . Hence Id P = p ◦ i = p ◦ (ε 1 i 1 + · · · + ε n i n ) : P → n P i → P∐and then P = n P i so that P is finitely generated.ε F ii∈1Assume now that P is finitely generated. Let us denote by ε j : B j → ∐ B i ,i∈I: B i → ∐ B i the canonical injections for every F ⊆ I finite subset, and let usprove thati∈F∇ (Hom (P, ε i )) i∈I: ∐ i∈IHom A (P, B i ) −→ Hom A(P, ∐ i∈IB i)i∈1255(f i ) i∈I↦→ ∑ i∈Iε i ◦ f iis an isomorphism. First of all we prove that it is epi. Since P is finitely generated, iff : P → ∐ i∈IB i is a morphism in A, by Lemma A.16, there exists a F ⊆ I finite subsetsuch that Im (f) ⊆ ∑ ε i (B i ) . Let us denote by f : P → Im (f) and s : Im (f) ↩→∑i∈Fε i (B i ) the canonical inclusion. Let us consider h = ∇ (ε i ) i∈F: ∐ B i → ∐ B ii∈Fi∈F i∈Isatisfying(265) h ◦ ε F i = ∇ (ε i ) i∈F◦ ε F i = ε i for every i ∈ F .Then, by definition of the codiagonal morphism, we have that Im (h) = ∑ ε i (B i )i∈Fand thus, if we call ĥ : ∐ B i → Im (h) = ∑ ε i (B i ) the canonical projection, wei∈F i∈Fcan write(266) h = t ◦ hwhere t : ∑ ε i (B i ) → ∐ B i is the inclusion. Thus also f can be factorized throughi∈Fi∈If by(267) f = t ◦ s ◦ f.With these notations we can rewrite (265) as follows(268) ε i = h ◦ ε F i = t ◦ h ◦ ε F i .