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

Abstract

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