ar X iv : m at h / 06 04 41 2 v 3 [ m at h . C T ] 4 S ep 2 00 6 UNIVERSAL COEFFICIENT THEOREM IN TRIANGULATED CATEGORIES

We consider a homology theory h : T → A on a triangulated category T with values in a graded abelian category A . If the functor h reflects isomorphisms, is full and is such that for any object x in A there is an object X in T with an isomorphism between h(X) and x, we prove that A is a hereditary abelian category, all idempotents in T split and the kernel of h is a square zero ideal which as a bifunctor on T is isomorphic to Ext 1 A (h(−)[1], h(−)). We assume that the reader is familiar with triangulated categories (see [7], [4]). Let us just recall that the triangulated categories were introduced independently by Puppe [6] and by Verdier [7]. Following to Puppe we do not assume that the octahedral axiom holds. If T is a triangulated category, the shifting of an object X ∈ T is denoted by X [1]. Assume an abelian category A is given, which is equipped with an autoequivalence x 7→ x[1]. Objects of A are denoted by the small letters x, y, z, etc, while objects of T are denoted by the capital lettersX,Y, Z, etc. A homology theory on T with values in A is a functor h : T → A such that h commutes with shifting (up to an equivalence) and for any distinguished triangle X → Y → Z → X [1] in T the induced sequence h(X) → h(Y ) → h(Z) is exact. It follows that then one has the following long exact sequence · · · → h(Z)[−1] → h(X) → h(Y ) → h(Z) → h(X)[1] → · · · In what follows ExtA (x, y) denotes the equivalence classes of extensions of x by y in the category A and we assume that these classes form a set. In this paper we prove the following result: Theorem 1. Let h : T → A be a homology theory. Assume the following conditions hold i) h reflects isomorphisms, ii) h is full. Then the ideal I = {f ∈ HomT (X,Y ) | h(f) = 0} is a square zero ideal. Suppose additionally the following condition holds iii) for any short exact sequence 0 → x → y → z → 0 in A with x ∼= h(X) and z ∼= h(Z) there is an object Y ∈ T and an isomorphism h(Y ) ∼= y in A . 2000 Mathematics Subject Classification. 18E30. The second author is a researcher from CONICET, Argentina. 1 2 TEIMURAZ PIRASHVILI AND MARÍA JULIA REDONDO Then I is isomorphic as a bifunctor on T to (X,Y ) 7→ ExtA (h(X)[1], h(Y )). In particular for any X,Y ∈ T one has the following short exact sequence 0 → ExtA (h(X)[1], h(Y )) → T (X,Y ) → HomA (h(X), h(Y )) → 0. Moreover, if we replace condition (iii) by the stronger condition iv) for any object x ∈ A there is an object X ∈ T and an isomorphism h(X) ∼= x in A , then A is a hereditary abelian category and all idempotents in T split. Thus this is a sort of ”universal coefficient theorem” in triangulated categories. Our result is a one step generalization of a well-known result which claims that if h is an equivalence of categories then A is semi-simple meaning that ExtA = 0 (see for example [4, p. 250]). As was pointed out by J. Daniel Christensen our theorem generalizes Theorem 1.2 and Theorem 1.3 of [3] on phantom maps. Indeed let S be the homotopy category of spectra or, more generally, a triangulated category satisfying axioms 2.1 of [3] and let A be the category of additive functors from finite objects of S to the category of abelian groups. The category A has a shifting, which is given by (F [1])(X) = F (X [1]), F ∈ A . Moreover let h : S → A be a functor given by h(X) = π0(X ∧ (−)). Then h is a homology theory for which the assertions i)-iii) hold and I(X,Y ) consists of phantom maps from X to Y . Hence by the first part of theorem we obtain the familiar properties of phantom maps. Before we give a proof of the Theorem, let us explain notations involved on it. The functor h reflects isomorphisms, this means that f ∈ HomT (X,Y ) is an isomorphism provided h(f) is an isomorphism in A . This holds if and only if X = 0 as soon as h(X) = 0. Moreover h is full, this means that the homomorphism T (X,Y ) → HomA (h(X), h(Y )) given by f 7→ h(f) is surjective for all X,Y ∈ T . Furthermore an abelian category A is hereditary provided for any two-fold extension (1) 0 // u α̂ // v β̂ // w γ̂ // x // 0 there exists a commutative diagram with exact rows 0 // v //