Generalised Moore Spectra in a Triangulated Category

In this paper we consider a construction in an arbitrary triangulated category T which resembles the notion of a Moore spectrum in algebraic topology. Namely, given a compact object C of T satisfying some finite tilting assumptions, we obtain a functor which “approximates” objects from the module category of the endomorphism algebra of C in T . This generalises and extends a construction of Jørgensen which appears in [10] in connection with lifts of certain homological functors of derived categories. We show that this new functor is well-behaved with respect to short exact sequences and distinguished triangles, and as a consequence we obtain a new way of embedding a module category in a triangulated category. As an example of the theory, we recover Keller’s canonical embedding of the module category of a path algebra of a quiver with no oriented cycles into its u-cluster category for u > 2. Introduction In this paper we discuss the existence of “Moore spectra in a triangulated category. The terminology “Moore spectra” employed in this paper is borrowed from algebraic topology, see [14]. While the notion discussed here does not coincide with its counterpart in algebraic topology its spirit is the same. In algebraic topology, the notion of spectra can be considered as one of “generalised topological spaces”. In this setting one uses the idea of a Moore spectrum to construct a spectrum with a single (pre-defined) non-vanishing homology group; c.f. the notion of an Eilenberg-MacLane space for homotopy groups. For instance, suppose A is an abelian group, the Moore spectrum MA of A is a spectrum with H(ΣMA) = { A i = 0 0 i 6= 0, where Σ is the suspension functor in the category of spectra. Analogously, in this paper we shall consider the following setup: suppose C is a compact object of a triangulated category T which has set indexed coproducts satisfying some finite tilting assumptions (see Setup 3.3 for precise Date: 30th March 2009.