Hall Algebra in a Triangulated Category

By counting with triangles and the octohedral axiom, we find a direct way to construct the Ringel-Hall algebra associated to a triangulated category with (left) homological-finite condition. Our formula is a refinement of that in Peng-Xiao [5] and provides a modified version of Toën’s formula in [7]. 1. Calculation with triangles Given a finite field k with q elements, let C be a k-additive triangulated category with the translation T = [1]. We always assume in this paper that (1) the KrullSchmidt theorem holds in C, i.e., any object in C can be decomposed into the direct sum of finitely many indecomposable objects, (2) the homomorphism space HomC(X,Y ) for any two objects X and Y in C is a finite dimensional k-space, and (3) the endomorphism ring EndX for any indecomposable object X is finite dimensional local k-algebra. Moreover, we always assume that C is (left) locally homological finite, i.e., ∑ i≥0 dim kHom(X [i], Y ) < ∞ for any X and Y in C. We will use fg to denote the composition of morphisms f : X → Y and g : Y → Z, and |A| the cardinality of a finite set A. The following is an easy result in [4]. Lemma 1.1. Given a triangle of form (1) M (f1,f2) // N1 ⊕N2 