Category equivalences involving graded modules over weighted path algebras and weighted monomial algebras

Let k be a field, Q a finite directed graph, and kQ its path algebra. Make kQ an N-graded algebra by assigning each arrow a positive degree. Let I be an ideal in kQ generated by a finite number of paths and write A=kQ/I. Let QGr A denote the quotient of the category of graded right A-modules modulo the Serre subcategory consisting of those graded modules that are the sum of their finite dimensional submodules. This paper shows there is a finite directed graph Q′ with all its arrows placed in degree 1 and an equivalence of categories QGrA≡QGrkQ′. A result of Smith now implies that QGrA≡ModS, the category of right modules over an ultramatricial, hence von Neumann regular, algebra S.