On perfectly generating projective classes in triangulated categories

arXiv:0811.0404v2 [math.CT] 17 Mar 2009 ON PERFECTLY GENERATING PROJECTIVE CLASSES IN TRIANGULATED CATEGORIES GEORGE CIPRIAN MODOI Abstract. We say that a projective class in a triangulated category with coproducts is perfect if the corresponding ideal is closed under coproducts of maps. We study perfect projective classes and the associ- ated phantom and cellular towers. Given a perfect generating projective class, we show that every object is isomorphic to the homotopy colimit of a cellular tower associated to that object. Using this result and the Neeman’s Freyd–style representability theorem we give a new proof of Brown Representability Theorem. Introduction The notion of projective classes in pointed categories goes back to Eilen- berg and Moore [4]. In this paper we consider projective classes in a category T which is triangulated. In this settings projective classes may be defined as pairs (P, F), with P ⊆T a class of objects and F ⊆T →a class of maps (here T →is the category of all maps in T ) such that P is closed under direct factors, F is an ideal (that means φ, φ ∈F, and α, β ∈T →, im- plies φ + φ′, αφβ ∈F, whenever the operations are defined), the composite p →x φ→x′ is zero for all p ∈P and all φ ∈F, and each object x ∈T lies in an exact triangle Σ−1x′ →p →x φ→x′, with p ∈P and φ ∈F. Note also that all projective classes which we deal with are stable under suspensions and desuspensions in T . Fix an object x ∈T . Choosing repeatedly triangles as above, we construct two towers in T associated to x, namely the phantom and the cellular tower. The whole construction is similar to the choice of a projective resolution for an object in an abelian category. Let κ be a regular cardinal. We say that a projective class (P, F) is κ– perfect, provided that the ideal F is closed under κ–coproducts in T →, that is coproducts of less that κ maps, respectively perfect if it is κ-perfect for all cardinals κ. For projective classes which are induced by sets our definition of perfectness is equivalent to that of [10], explaining our terminology. Further we say that (P, F) generates T if for any x ∈T , T (P, x) = 0 implies x = 0. It seems that an important role is played by ℵ1–perfect projective classes, that 2000 Mathematics Subject Classification. 18E15, 18E35, 18E40, 16D90. Key words and phrases. triangulated category with coproducts, perfect projective class, Brown representability. The author was supported by the grant PN2CD-ID-489. 1 2 GEORGE CIPRIAN MODOI means projective classes (P, F) with F closed under countable coproducts. In this case we prove that the homotopy colimit of a tower whose maps belong to F is zero (see Lemma 2.2). In particular the homotopy colimit of the phantom tower associated to an object vanishes. If, in addition, we assume that (P, F) generates T then Theorem 2.5 tells us that every object x is (isomorphic to) the homotopy colimit of every associated cellular tower. Note also that the hypothesis of ℵ1–perfectness seems to be implicitly assumed by Christensen in [3], as we may see from Proposition 2.3 and Remark 2.4. Using the product of two projective classes defined in [3] we recall the construction of the n-th power (P∗n, F∗n) of a projective class (P, F), for n ∈N. In [14] it is shown that, if (P, F) is induced by a set, then for every cohomological functor F : T →Ab which sends coproducts into products the comma category P∗n/F has a weak terminal object, for all n ∈N. Provided that (P, F) is ℵ1–perfect, we use the fact that every x is the homotopy colimit of its cellular tower in order to extend the above property to the whole category T /F. We deduce a version of Brown Representability Theorem for triangulated categories with coproducts which are ℵ1-perfectly generated by a projective class satisfying the additional property that every category P∗n/F has a weak terminal object, for every n ∈N and every cohomological functor which sends coproducts into products F : T →Ab (see Theorem 3.7). In particular if the projective class is induced by a set, then this additional property is automatically fulfilled, and we obtain in Corollary 3.8 the version of Brown Representability due to Krause in [8, Theorem A], but our proof is completely different, as it is based on the Freyd–style representability theorem of [14]. For our version of Brown Representability the finite powers of a projective class is all what we need. We still treated the case of transfinite ordinals, following a suggestion of Neeman (see [14, Remark 0.10]). A minor modifi- cation of the arguments in [14] shows that if T = P∗i for some ordinal i then the Brown representability theorem holds for T . We fill in the details this observation in Lemmas 3.3 and 3.4. On the other hand, if every x ∈T is the homotopy colimit of its F–cellular tower, then T = P∗ω ∗P∗ω, where ω is the first infinite ordinal. But due to a technical detail we are not able to deduce, as in the case of finite ordinals (see [3, Note 3.6]), that P∗ω ∗P∗ω = P∗(ω+ω)

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