Rationality of the Poincaré series for Koszul algebras

In this paper, we study the rationality of the Poincaré series of a finitely generated graded left module over a Koszul connected algebra. In particular, we show the following result: let A be a Koszul connected algebra and A! be its Koszul dual. Suppose that A! is noetherian and having a balanced dualizing complex. If A is either (1) an artinian algebra, (2) a graded quotient algebra of a noetherian AS-regular algebra, or (3) an FBN algebra (e.g., a noetherian PI algebra), then the Poincaré series of every finitely generated graded left A-module is a rational function over the complex numbers.