From Green's formula to Derived Hall algebras

TL;DR

This paper demonstrates that Green's formula and the associativity of the derived Hall algebra are logically equivalent, clarifying their relationship in the context of finitary hereditary abelian categories.

Contribution

It proves that Green's formula implies the associativity of the derived Hall algebra, establishing a two-way equivalence.

Findings

Green's formula implies associativity of derived Hall algebra

Associativity of derived Hall algebra implies Green's formula

Clarifies the relationship between Green's formula and derived Hall algebra

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