From Green's formula to Derived Hall algebras

Abstract: The aim of this note is to clarify the relationship between Green's formula and the associativity of multiplication for derived Hall algebra in the sense of To\"{e}n (Duke Math J 135(3):587-615, 2006), Xiao and Xu (Duke Math J 143(2):357-373, 2008) and Xu and Chen (Algebr Represent Theory 16(3):673-687, 2013). Let $\mathcal{A}$ be a finitary hereditary abelian category. It is known that the associativity of derived Hall algebra $\mathcal{D}\mathcal{H}_t(\mathcal{A})$ implies Green's formula. We show the converse statement holds. Namely, Green's formula implies the associativity of the derived Hall algebra $\mathcal{D}\mathcal{H}_t(\mathcal{A})$.