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

Lukas Katthän, Julio José Moyano-Fernández, Jan Uliczka

TL;DR

This work develops a comprehensive framework to determine which formal Laurent series can arise as Hilbert series of finitely generated modules over multigraded polynomial rings with unit-vector variable degrees. It provides a complete $\mathbb{Z}^n$-graded classification, along with sharp results for the fine-graded and bigraded special cases, by establishing necessary and sufficient conditions based on Hilbert polynomials of restrictions, positive extremal coefficients, and explicit linear inequalities tied to depth (Hilbert depth). A key contribution is a family of general inequalities for Hilbert depth that links projective dimension, syzygies, and Hilbert series, plus explicit combinatorial criteria in the bigraded and non-standard graded settings (declining sequences and fundamental couples). The results illuminate the precise structure of Hilbert series in multigraded contexts, clarify obstructions to naive positivity, and extend classical single-graded characterizations to richer multigraded environments with concrete, testable conditions. These findings have implications for understanding growth, depth, and decomposition properties of multigraded modules and their Hilbert series representations.

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.

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

TL;DR

This work develops a comprehensive framework to determine which formal Laurent series can arise as Hilbert series of finitely generated modules over multigraded polynomial rings with unit-vector variable degrees. It provides a complete -graded classification, along with sharp results for the fine-graded and bigraded special cases, by establishing necessary and sufficient conditions based on Hilbert polynomials of restrictions, positive extremal coefficients, and explicit linear inequalities tied to depth (Hilbert depth). A key contribution is a family of general inequalities for Hilbert depth that links projective dimension, syzygies, and Hilbert series, plus explicit combinatorial criteria in the bigraded and non-standard graded settings (declining sequences and fundamental couples). The results illuminate the precise structure of Hilbert series in multigraded contexts, clarify obstructions to naive positivity, and extend classical single-graded characterizations to richer multigraded environments with concrete, testable conditions. These findings have implications for understanding growth, depth, and decomposition properties of multigraded modules and their Hilbert series representations.

Abstract

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

Paper Structure

This paper contains 11 sections, 18 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Notation and preliminaries
  4. Multivariate Hilbert Polynomials
  5. Which series are Hilbert series?
  6. The standard $\mathbb{Z}^n$-grading
  7. Non-Negative series
  8. The fine graded case
  9. General inequalities for the Hilbert depth
  10. The Bigraded case
  11. The non-standard Z-graded case

Key Result

Theorem 1.1

Let $R = \bigoplus_\mathbf{a} R_\mathbf{a}$ be a (commutative Noetherian) $\mathbb{Z}^n$-graded $\mathbb{K}$-algebra, such that $\dim_\mathbb{K} R_0 < \infty$. Let further $p \in \mathbb{N}$ and $M, N$ be finitely generated $R$-modules. If $\mathop{\mathrm{pdim}}\nolimits M \leq p$ and $N$ is a $p$-

Figures (1)

  • Figure 1: A declining sequence $U=\left\{(1,5),(3,4),(4,3),(7,1)\right\}$ with $\mathcal{D}(U)=\left\{(1,4) ,(3,3),(4,1)\right\}$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Theorem 3.1
  • Example 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.5
  • Remark 3.6
  • ...and 35 more