From Green's formula to Derived Hall algebras

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 Toën (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 $\mathcal{D}{\mathcal{H}}_t(\mathcal{A})$.