Derived Hall algebras

The purpose of this work is to define a derived Hall algebra DH(T), associated to any differential graded (DG) category T (under some finiteness conditions), generalizing the Hall algebra of an abelian category. Our main theorem states that DH(T) is associative and unital. When the associated triangulated category [T] is endowed with a t-structure with heart A, it is shown that DH(T) contains the usual Hall algebra H(A).We also prove an explicit formula for the derived Hall numbers purely in terms of invariants of the triangulated category associated to T. As an example, we describe the derived Hall algebra of a hereditary abelian category.