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 of the module category of the endomorphism algebra of C in T. This generalises and extends a construction of Jorgensen 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 the 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>1.

💡 Research Summary

The paper develops a construction that mimics the classical Moore spectrum, but inside an arbitrary triangulated category T. The starting data is a compact object C in T that satisfies a finite‑tilting condition: roughly, C generates T through a finite number of copies, which guarantees that the endomorphism algebra A = End_T(C) is a finite‑dimensional k‑algebra. Under these hypotheses the authors define a functor

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