BROWN REPRESENTABILITY AND FLAT COVERS

HENNING KRAUSE

Abstract. 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 F^{op} → 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

H_{X} = Hom(−, X)|_{F}: F^{op} ⟶ Ab.

This gives rise to a functor T → (F^{op}, Ab) into the category of additive functors F^{op} → Ab. Every functor of the form H_{X} 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 F^{op} → Ab is isomorphic to H_{X} for some object X in T , and

• every natural transformation H_{X} → H_{Y} is of the form H_{f} 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 (F^{op}, Ab). Recall that there exists a tensor product

(F^{op}, Ab) × (F, Ab) ⟶ Ab, (F, G) → F ⊗_{F} G

where for any functor F: F^{op} → 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: F^{op} → 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

• E is flat and every map E′ → F such that E′ is flat factors through p;

• every endomorphism e: E → E satisfying pe = p is an isomorphism. This is a useful concept because flat covers usually exist whereas projective covers rarely exist. In fact, Enochs conjectured in [5] that every module over an associative ring has a flat cover. Note that a module category is a special case of a functor category. In any case, Enochs’ conjecture seems to be hard; it is still open [9]. Let us construct for every functor F: F^{op} → Ab an object X ∈ T and a map p_{F} : H_{X} → F which is a good candidate for being a flat cover of F. Choose a minimal injective copresentation of the form

0 ⟶ F ⟶ H_{I} ⟶^{H_{f}} H_{J}

in (F^{op}, Ab). This is possible by Lemma 1 below. The map f can be completed to a triangle

X ⟶^{e} I ⟶^{f} J ⟶^{g} ΣX,

and the map H_{e}: H_{X} → H_{I} induces a map p_{F} : H_{X} → F. Note that p_{F} is uniquely determined by F, up to isomorphism. We are now in a position to state the main result of this note.

Theorem. Let T be a compactly generated triangulated category and denote by F the full subcategory of compact objects in T . Then the following conditions are equivalent:

(1) Brown representability holds for T .

(2) Every exact functor F^{op} → Ab is isomorphic to H_{X} for some object X in T , and every isomorphism H_{X} → H_{Y} is of the form H_{f} for some map f: X → Y in T .

(3) A functor F: F^{op} → Ab is flat if and only if Ext^{2}(−, F) = 0.

(4) For every functor F: F^{op} → Ab the map p_{F} : H_{X} → F is a flat cover of F.

For a local version of this result we refer to the proposition at the end of this note. The theorem is based on the interplay between the triangulated structure of T and the abelian structure of the functor category (F^{op}, Ab). A similar approach can be found in recent work of Christensen, Strickland, and Neeman [4, 8]. In fact, the connection between Brown representability and the vanishing of Ext^{2}(−, F) for exact functors F has been noticed by these authors. The new ingredient which leads to the connection with flat covers, is the existence of injective envelopes in (F^{op}, Ab), see [6]. Recall that a monomorphism i: F → Q in any abelian category is an injective envelope of F, if Q is injective and every endomorphism e: Q → Q satisfying ei = i is an isomorphism.

The rest of this paper is devoted to proving the main result. To this end we need to recall our assumptions on the triangulated category T :

• T has arbitrary coproducts;

• the isomorphism classes of compact objects in T form a set;

• H_{X} = 0 implies X = 0 for every object X in T . The proof of the theorem is divided into several lemmas. The first one is taken from [7]. We include the short proof for the convenience of the reader.

Lemma 1. Let Q be an injective object in (F^{op}, Ab). Then there exists, up to isomorphism, a unique object I in T such that Q ≅ H_{I}. Moreover, the map

Hom(X, I) ⟶ Hom(H_{X}, H_{I}), f ↦ H_{f},

is an isomorphism for all X in T .

Proof. Consider the exact functor F: T^{op} → Ab, X ↦ Hom(H_{X}, Q). It sends coproducts in T to products in Ab and is therefore representable by Brown’s classical theorem [2]. Thus F ≅ Hom(−, I) for some object I in T . The induced map Hom(I, I) ≅ F (I) = Hom(H_{I}, Q) sends id_{I} to a map p: H_{I} → Q which is an isomorphism since

H_{I}(C) = Hom(C, I) ≅ Hom(H_{C}, Q) ≅ Q(C)

for every compact object C by Yoneda’s lemma. The inverse p^{−1} : Q → H_{I} induces an isomorphism

Hom(X, I) ≅ Hom(H_{X}, Q) ≅ Hom(H_{X}, H_{I})

which is precisely the map f ↦ H_{f}. This finishes the proof.

Lemma 2. Let F ∈ (F^{op}, Ab). Then Ext^{2}(−, F) = 0 if and only if the canonical map p_{F} : H_{X} → F is an isomorphism.

Proof. We fix the minimal injective copresentation

0 ⟶ F ⟶ H_{I} ⟶^{H_{f}} H_{J}

and denote by e: X → I the fiber of f: I → J. We have Ext^{2}(−, F) = 0 if and only if H_{f} is an epimorphism, and this is equivalent to H_{e} being a monomorphism since the fiber e gives rise to an exact sequence

H_{Σ^{−1}J} ⟶^{H_{Σ^{−1}g}} H_{X} ⟶^{H_{e}} H_{I} ⟶^{H_{f}} H_{J} ⟶^{H_{g}} H_{ΣX}.

Furthermore, H_{e} is a monomorphism if and only if H_{X} is a kernel of H_{f}. We conclude from the construction of the map p_{F} that Ext^{2}(−, F) = 0 if and only if p_{F} is an isomorphism.

Lemma 3. Let X →^{e} I →^{f} J →^{g} ΣX be a triangle in T and suppose that the induced map Im H_{f} → H_{J} is an injective envelope. Then every endomorphism u: X → X satisfying eu = e is an isomorphism.

Proof. Choose a map v: J → J which completes the following commutative diagram

X ⟶^{e} I ⟶^{f} J ⟶^{g} ΣX
↓u  ∥  ↓Σu
X ⟶^{e} I ⟶^{f} J ⟶^{g} ΣX

The assumption on u and f imply that H_{v} is an isomorphism. Thus v is an isomorphism, and we conclude that u is an isomorphism.

Proposition. The following conditions are equivalent for an object Y in T :

(1) Ext^{2}(−, H_{Y}) = 0.

(2) Every map H_{X} → H_{Y} is of the form H_{u} for some map u: X → Y.

Proof. (1) ⇒ (2) Fix a map H_{X} → H_{Y}. We use Lemma 1 several times. First we choose an injective envelope H_{e}: H_{Y} → H_{I} and complete the map e to a triangle

Y ⟶^{e} I ⟶^{f} J ⟶^{g} ΣY

where H_{J} is injective by our assumption on X. The composite H_{X} → H_{Y} → H_{I} must come from a map X → I. The composite X → I → J vanishes. Hence there is a factorization X → Y → I, and the map X → Y induces the map H_{X} → H_{Y} by construction.

(2) ⇒ (1) We fix a minimal injective copresentation

0 ⟶ H_{Y} ⟶ H_{I} ⟶^{H_{f}} H_{J}

and complete f to a triangle

X ⟶^{e} I ⟶^{f} J ⟶^{g} ΣX.

By (2), the induced map p: H_{X} → H_{Y} is of the form p = H_{v}. The map c: Y → I which induces H_{Y} → H_{I} factors through e via some map w: Y → X since fc = 0. The composite wv is an endomorphism of X satisfying ewv = e. Therefore wv is an isomorphism by Lemma 3. In particular p = H_{v} is a monomorphism. On the other hand, p is an epimorphism since pH_{w} = id_{H_{Y}}. It follows from Lemma 2 that Ext^{2}(−, H_{Y}) = 0.

Proof of the Theorem. Throughout the proof we shall use freely the fact that flat and exact functors in (F^{op}, Ab) coincide (see Lemma 2.7 in [7]).

(1) ⇔ (2) A map f: H_{X} → H_{Y} is of the form H_{u} for some u: X → Y if and only if the isomorphism

[ \begin{matrix}id_{H_{X}} & f \\ 0 & id_{H_{Y}} \end{matrix} ] : H_{X} ∐ H_{Y} ⟶ H_{X} ∐ H_{Y}

is of the form H_{v} for some v: X ∐ Y → X ∐ Y.

(1) ⇒ (3) If Ext^{2}(−, F) = 0, then F is exact by Lemma 2 and therefore flat. Conversely, suppose that F is flat and therefore exact. Brown representability implies F ≅ H_{Y} for some Y ∈ T , and the representability of morphisms implies Ext^{2}(−, F) = 0 by the Proposition.

(3) ⇒ (4) Let F ∈ (F^{op}, Ab), and suppose that E → F is any map such that E is flat. We need to show that E → F factors through p_{F}. We take the composite

H_{Y} ⟶^{p_{E}} E ⟶ F ⟶ H_{I}

which is of the form H_{c} for some map c: Y → I by Lemma 1. Clearly, H_{f}H_{c} = H_{fc} = 0, and therefore fc = 0, again by Lemma 1. Therefore c factors through e: X → I via some map u: Y → X. Condition (3) and Lemma 2 imply that p_{E} is an isomorphism, and the composite H_{u}p_{E}^{−1} : E → H_{X} gives the desired map.

Now let v: H_{X} → H_{X} be any endomorphism satisfying p_{F} v = p_{F}. We need to show that v is an isomorphism. To this end observe that v = H_{u} for some u: X → X by the Proposition. Moreover, p_{F} v = p_{F} implies H_{eu} = H_{e}, and therefore eu = e by Lemma 1. An application of Lemma 3 shows that v is an isomorphism, and this completes the proof of the fact that p_{F} is a flat cover of F.

(4) ⇒ (1) Clearly, a flat cover E → F is an isomorphism if F is flat. Therefore (4) implies that P_{F} : H_{X} → F is an isomorphism for every exact functor F. A combination of Lemma 2 and the Proposition shows that every map H_{X} → H_{Y} is induced by a map X → Y in T .

Acknowledgement. This paper was written in Spring 1999 while I visited Santa Barbara. I would like to thank Birge Huisgen-Zimmermann and Ken Goodearl for their warm hospitality. Also, I am indebted to Dan Christensen for a number of suggestions which helped to simplify some of my arguments.

References

[1] J.F. Adams, A variant of E.H. Brown’s representability theorem, Topology 10 (1971), 185–198.

[2] E.H. Brown, Cohomology theories, Ann. of Math. 75 (1962), 467–484.

[3] D.J. Benson and G.P. Gnacadja, Phantom maps and purity in modular representation theory I, Fund. Math., to appear.

[4] J.D. Christensen and N.P. Strickland, Phantom maps and homology theories, Topology 37 (1998), 339–364.

[5] E.E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189–209.

[6] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.

[7] H. Krause, Smashing subcategories and the telescope conjecture - an algebraic approach, Invent. Math., to appear.

[8] A. Neeman, On a theorem of Brown and Adams, Topology 36 (1997), 619–645.

[9] J. Xu, Flat covers of modules, Springer Lec. Notes 1634 (1996).

Department of Mathematics, University of California, Santa Barbara, CA 93106, U.S.A.

E-mail address: krause@math.ucsb.edu

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

E-mail address: henning@mathematik.uni-bielefeld.de