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 quasiisomorphic 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.Let A be a finite dimensional algebra over a field k. The homotopy category K ac (A-Inj) of acyclic complexes of injective A-modules is called the stable derived category of A in [12]. This category is a compactly generated triangulated category such that its subcategory of compact objects is triangle equivalent to the singularity category [5,15] of A.In general, it seems very difficult to give an explicit compact generator for the stable derived category of an algebra. In this paper, we construct an explicit compact generator for the homotopy category K ac (A-Inj), in the case that the algebra A is with radical square zero. The compact generator is called the injective Leavitt complex. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex is quasi-isomorphic to the Leavitt path algebra, which was introduced in [2, 3] as an algebraisation of graph C * -algebras [13,16] and in particular Cuntz-Krieger algebras [10]. Here, the Leavitt path algebra is naturally Z-graded and viewed as a differential graded algebra with trivial differential.Let Q be a finite quiver without sinks. Set A = kQ/J 2 to be the corresponding finite dimensional algebra with radical square zero. We introduce the injective Leavitt complex I • in Definition 1.4, which is an acyclic complex of injective A-modules; see Proposition 1.9.We denote by L k (Q) the Leavitt path algebra of Q over k. Recall that L k (Q) is a naturally Z-graded algebra, which is viewed as a differential graded algebra with trivial differential.The main result of this paper is as follows, which combines Theorem 2.13 with Theorem 4.2.Theorem I. Let Q be a finite quiver without sinks and A = kQ/J 2 the corresponding algebra with radical square zero.(1) The injective Leavitt complex I • of Q is a compact generator for the homotopy category K ac (A-Inj).(2) The differential graded endomorphism algebra of the injective Leavitt complexThis result is inspired by [9, Theorem 6.1], which describes the homotopy category K ac (A-Inj) in terms of Leavitt path algebras. We mention that [9, Theorem 6.1] extends [17, Theorem 7.2]; compare [7, Theorem 3.8]. The construction of the injective Leavitt complex is inspired by the basis of the Leavitt path algebra given by [1, Theorem 1].For the proof of (1) in Theorem I, we construct an explicit filtration of subcomplexes of the injective Leavitt complex I • . For (2)...

The Projective Leavitt Complex

“…We have the following observation. The proof of it is similar as that of [12,Lemma 2.7]. We omit it here.…”

“…Observe that for each vertex i and each integer l, Λ l i is one-to-one corresponded to {(q op , p op ) | (q op , p op ) is an admissible pair in Q op with l(p op ) − l(q op ) = −l and s(p op ) = i}. Here, refer to [12,Definition 2.1] for the definition of admissible pair. By [12,Lemma 2.2], the later set is not empty.…”

“…Here, refer to [12,Definition 2.1] for the definition of admissible pair. By [12,Lemma 2.2], the later set is not empty. The proof is completed.…”

“…Observe that A is a self-injective algebra. The projective Leavitt complex P • is isomorphic to the injective Leavitt complex I • as complexes; compare [12,Example 2.11].…”