The Projective Leavitt Complex

Abstract

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

The Injective and Projective Leavitt Complexes

“…We overview the connection between the injective and projective Leavitt complex and the Leavitt path algebra of the given graph. A differential graded bimodule structure, which is right quasi-balanced, is endowed to the injective and projective Leavitt complex in [18] and [19]. We prove that the injective and projective Leavitt complex is not left quasi-balanced.…”

“…2) The set Λ l i is not empty for each vertex i and each integer l; see [19,Lemma 2.2]. Recall that A = kE/kE ≥2 is a finite dimensional algebra with radical square zero.…”

“…Each component P l is a projective A-module and the differential δ l are A-module morphisms. P • is an acyclic complex of injective A-modules; see [19,Proposition 2.7]. Example 4.3.…”

“…The projective Leavitt complex P • = l∈Z P l is a left dg A-module. By [19,Proposition 4.6], P • is a right dg B-module and a dg A-B-bimodule.…”

“…This category is a compactly generated triangulated category whose subcategory of compact objects is triangle equivalent to the opposite category of the singularity category of the opposite algebra A op . An explicit compact generator, called the projective Leavitt complex, for the homotopy category K ac (A-Proj) in the case that A is an algebra with radical square zero was constructed in [19]. It is show that the opposite differential graded endomorphism algebra of the projective Leavitt complex of a finite quiver without sources is quasi-isomorphic to the Leavitt path algebra of the opposite graph [19].…”