Brown Representability and Flat Covers

We exhibit a surprising connection between the following two concepts: Brown representability which arises in stable homotopy theory, and flat covers which arise in module theory. It is shown that Brown representability holds for a compactly generated triangulated category if and only if for every additive functor from the category of compact objects into the category of abelian groups a flat cover can be constructed in a canonical way. The proof also shows that Brown representability for objects and morphisms is a consequence of Brown representability for objects and isomorphisms. In this note we exhibit a surprising connection between the following two concepts: • Brown representability which arises in stable homotopy theory, and • flat covers which arise in module theory. Let T be a compactly generated triangulated category, for example the stable homotopy category of CW-spectra, and denote by F the full subcategory of compact objects (an object X in T is compact if the representable functor Hom(X,−) preserves arbitrary coproducts). We prove that Brown representability holds for T if and only if for every additive functor Fop → Ab a flat cover can be constructed in a canonical way. The proof also shows that Brown representability for objects and morphisms is a consequence of Brown representability for objects and isomorphisms. Let us recall the relevant definitions. For every object X in T consider the functor HX = Hom(−,X)|F : F −→ Ab. This gives rise to a functor T → (Fop,Ab) into the category of additive functors Fop → Ab. Every functor of the form HX is exact, that is, it sends triangles in F to exact sequences in Ab. In some cases also the converse is true. More precisely, one says that Brown representability holds for T , if • every exact functor Fop → Ab is isomorphic to HX for some object X in T , and • every natural transformation HX → HY is of the form Hf for some map f : X → Y . A classical theorem due to Brown and Adams states that Brown representability holds for the stable homotopy category [1]. This has been generalized in recent work by Neeman, which also contains an example of a compactly generated triangulated category where Brown representability fails [8]. For the relevance of Brown representability in the representation theory of finite groups we refer to recent work of Benson and Gnacadja [3]. Now consider the functor category (Fop,Ab). Recall that there exists a tensor product (F,Ab)× (F ,Ab) −→ Ab, (F,G)→ F ⊗F G where for any functor F : Fop → Ab, the tensor functor F ⊗F − is determined by the fact that it preserves colimits and F ⊗F Hom(X,−) ∼= F (X) for all X in F . A functor F : Fop → Ab is flat if the tensor functor F ⊗F − is exact. Our interest in flat functors is motivated by the elementary fact that a functor is flat if and only if it is exact [7]. A map p : E → F between functors is called a flat cover of F , if