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