The injective Leavitt complex

Abstract

For a finite quiver $Q$ without sinks, we consider the corresponding finite dimensional algebra $A$ with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective $A$-modules. We call such a generator the injective Leavitt complex of $Q$. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of $Q$ is quasi-isomorphic to the Leavitt path algebra of $Q$. Here, the Leavitt path algebra is naturally Z-graded and viewed as a differential graded algebra with trivial differential.