Universal coe cient 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 re ects 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 Ext1A (h( )[1]; h( )). 2000 Mathematics Subject Classi cation: 18E30 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 2 T is denoted by X[1]. Assume an abelian category A is given, which is equipped with an auto-equivalence 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 letters X;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 re ects isomorphisms, ii) h is full. The second author is a researcher from CONICET, Argentina c 2008 Kluwer Academic Publishers. Printed in the Netherlands. ucoefK.tex; 6/05/2008; 18:04; p.1 2 Then the ideal I = ff 2 HomT (X;Y ) j h(f) = 0g 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 2 T and an isomorphism h(Y ) = y in A . 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 2 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 2 A there is an object X 2 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 coe cient 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 nite 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 2 A . Moreover let h : S ! A be a functor given by h(X) = 0(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 rst 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 re ects isomorphisms, this means that f 2 HomT (X;Y ) is an isomorphism provided h(f) is an isomorphism in A . ucoefK.tex; 6/05/2008; 18:04; p.2 3 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 2 T . Furthermore an abelian category A is hereditary provided for any two-fold extension 0 // u ̂ // v ̂ // w ̂ // x // 0 (1) there exists a commutative diagram with exact rows 0 // v //