# Hall algebras associated to complexes of fixed size

## TL;DR

This paper investigates the structure of Hall algebras formed from complexes of fixed size over projectives in a hereditary abelian category, establishing relations with cyclic complexes, derived Hall algebras, and providing an integration map.

## Contribution

It introduces a detailed description of Hall algebras of fixed-size complexes, relating them to cyclic and derived Hall algebras, and constructs an explicit integration map.

## Key findings

- Relation between Hall algebras of fixed size and cyclic complexes
- Characterization of Hall algebra via generators and relations
- Explicit integration map for 2-term complexes

## Abstract

Let $\A$ 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 $\A$. Finally, we give the integration map on the Hall algebra of $2$-term complexes over projectives.