Hall algebras associated to complexes of fixed size

Let $\text{A}$ be a finitary hereditary abelian category with enough projectives. We study the Hall algebra of complexes of fixed size over projectives. Explicitly, we first give a relation between Hall algebras of complexes of fixed size and cyclic complexes. Secondly, we characterize the Hall algebra of complexes of fixed size by generators and relations, and relate it to the derived Hall algebra of $\text{A}$. Finally, we give the integration map on the Hall algebra of $2$-term complexes over projectives.