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

248where k : Ker (Coker (f ◦ p)) → Y is the canonical monomorphism. Since p is anepimorphism, from formula (264) we obtain that f = k ◦ f ◦ p and hence f is amonomorphism.□Theorem A.9 ([Po, page 112]). Let A be a Grothendieck category, let U be an objectof A and let B = Hom A (U, U). Assume that U is a generator of A and that thereexists a left adjoint T : Mod-B → A of the functor Hom A (U, −) : A → Mod-B.Then T is an exact functor.Proof. By Proposition A.3, H is full and faithful. Since T is a left adjoint so that itpreserves epimorphisms, we have only to prove that it is left exact.Now let X be a submodule of a free right B-module B (Z) . Let P 0 (X) be the set offinite subset of X and let j F : X F → X be the canonical inclusion of the submoduleof X spanned by F ∈ P 0 (X). By Lemma A.7, for every F ∈ P 0 (X) there exists afinite subset Z F of Z and a monomorphism i F : X F → A (Z F ) such that the 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(h F ◦ i F ) F ∈P0 (X)is a compatible family of morphisms such thatlim (h F ◦ i F ) = j.−→Since T is a left adjoint functor, we have()T (j) = T lim (h F ◦ i F )−→= lim−→T (h F ◦ i F ) .By Lemma A.6 we know that T (i F ) is a monomorphism. On the other hand π F ◦h F = Id A (Z F ) and hence also T (h F ) is a monomorphism. Since A is a Grothendieckcategory, direct limits are exact in A and hence T (j) is a monomorphism.Finally let0 → L f−→ Mbe a monomorphism in Mod-B. Then we can construct the following commutativediagram with exact rows and columns000 Ker (p)i ′Pp ′L0Id Ker(p)0 Ker (p)f ′ i B (M) pf M 0