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 the derived Hall algebra \(\mathcal {D}\mathcal {H}_t(\mathcal {A})\) implies Green’s formula. We introduce a new algebra \({\mathcal {L}}_t({\mathcal {A}})\) whose associativity is deduced from Green’s formula, and show that it is isomorphic to the derived Hall algebra \(\mathcal {D}\mathcal {H}_t(\mathcal {A})\) .
