Which series are Hilbert series of graded modules over standard multigraded polynomial rings?

Abstract

Consider a polynomial ring.. with the Z(n)-grading where the degree of each variable is a standard basis vector. In other words, R is the homogeneous coordinate ring of a product of.. projective spaces. In this setting, we characterize the formal Laurent series which arise as Hilbert series of finitely generated R-modules. We also provide necessary conditions for a formal Laurent series to be the Hilbert series of a finitely generated module with a given depth. In the bigraded case (corresponding to the product of two projective spaces), we completely classify the Hilbert series of finitely generated modules of positive depth.