Module theory over Leavitt path algebras and [formula omitted]-theory

Abstract

Let k be a field and let E be a finite quiver. We study the structure of the finitely presented modules of finite length over the Leavitt path algebra L k ( E ) and show its close relationship with the finite-dimensional representations of the inverse quiver E ¯ of E , as well as with the class of finitely generated P k ( E ) -modules M such that Tor q P k ( E ) ( k | E 0 | , M ) = 0 for all q , where P k ( E ) is the usual path algebra of E . By using these results we compute the higher K -theory of the von Neumann regular algebra Q k ( E ) = L k ( E ) Σ − 1 , where Σ is the set of all square matrices over P k ( E ) which are sent to invertible matrices by the augmentation map ϵ : P k ( E ) → k | E 0 | .