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

TL;DR

This paper characterizes which formal Laurent series can be Hilbert series of finitely generated modules over multigraded polynomial rings, especially focusing on the bigraded case related to products of projective spaces.

Contribution

It provides a complete classification of Hilbert series for modules over multigraded polynomial rings, including necessary conditions and a full classification in the bigraded case.

Findings

Characterization of Hilbert series in multigraded settings

Necessary conditions for Hilbert series with given depth

Complete classification of Hilbert series in the bigraded case

Abstract

Let $S$ be a multigraded polynomial ring such that the degree of each variable is a unit vector; so $S$ 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 $S$-modules. Also we 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.