An equivalence of categories for graded modules over monomial algebras and path algebras of quivers

Let A be a finitely generated connected graded k-algebra defined by a finite number of monomial relations, or, more generally, the path algebra of a finite quiver modulo a finite number of relations of the form “path=0”. Then there is a finite directed graph, Q, the Ufnarovskii graph of A, for which there is an equivalence of categories . Here is the quotient category of graded A-modules modulo the subcategory consisting of those that are the sum of their finite dimensional submodules. The proof makes use of an algebra homomorphism A→kQ that may be of independent interest.