Triangulated Categories Part III

1 Brown Representability First we introduce a portly abelian category A(S) for any preadditive category S. This will be a certain full subcategory of the category of all contravariant additive functors S −→ Ab. Modulo the fact that S is not assumed to be small, this is precisely what we call a right module over a ringoid in our Rings With Several Objects (RSO) notes. For background on portly abelian categories, the reader is referred to (AC,Section 2.4). Definition 1. Given a preadditive categoryA, the objects of the portly abelian categoryModA = (A,Ab) are called right A-modules. A sequence of right modules M ′ −→M −→M ′′ is exact in ModA if and only if the following sequence is exact in Ab for every A ∈ A M ′(A) −→M(A) −→M ′′(A) Similarly kernels, cokernels and images in ModA are computed pointwise. See (AC,Corollary 59) and the proof of (AC,Proposition 44) for details. A morphism φ : M −→ N in ModA is a monomorphism or epimorphism if and only if φA : M(A) −→ N(A) has this property for every A ∈ A. For any object A ∈ A we have the right module HA = Hom(−, A) : A −→ Ab defined in the obvious way. Proposition 1 (Yoneda). If A is a preadditive category, then (i) For any object A ∈ A and right A-module T there is a canonical isomorphism of abelian groups HomA(HA, T ) −→ T (A) defined by γ 7→ γA(1). (ii) The functor A 7→ HA defines a full additive embedding A −→ModA. (iii) The objects {HA}A∈A form a (large) generating family of projectives for ModA.