Extensions of simple modules over Leavitt path algebras

Let E be a directed graph, K any field, and let L-K(E) denote the Leavitt path algebra of E with coefficients in K. For each rational infinite path c(infinity) of E we explicitly construct a projective resolution of the corresponding Chen simple left L-K(E)-module V-[c infinity] Further, when E is row-finite, for each irrational infinite path p of E we explicitly construct a projective resolution of the corresponding Chen simple left L-K(E)-module V-[p]. For Chen simple modules S,T we describe Ext(Lk(E)(1))S,T) by presenting an explicit K-basis. For any graph E containing at least one cycle, this description guarantees the existence of indecomposable left L-K(E)-modules of any prescribed finite length. (C) 2015 Elsevier Inc. All rights reserved.