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Hilbert polynomial of length functions

Antongiulio Fornasiero

Abstract

Let $λ$ be a general length function for modules over a Noetherian ring R. We use $λ$ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of~$λ$. We show that the leading term $μ$ of the Hilbert polynomial is an invariant of the module, which refines both the algebraic entropy and the receptive algebraic entropy; its degree is a suitable notion of dimension for $R[X]$-modules. Similar to algebraic entropy, $μ$ in general is not additive for exact sequence of $R[X]$-modules: we demonstrate how to adapt of certain entropy constructions to this new invariant. We also consider multi-variate versions of the Hilbert polynomial.

Hilbert polynomial of length functions

Abstract

Let be a general length function for modules over a Noetherian ring R. We use to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of~. We show that the leading term of the Hilbert polynomial is an invariant of the module, which refines both the algebraic entropy and the receptive algebraic entropy; its degree is a suitable notion of dimension for -modules. Similar to algebraic entropy, in general is not additive for exact sequence of -modules: we demonstrate how to adapt of certain entropy constructions to this new invariant. We also consider multi-variate versions of the Hilbert polynomial.

Paper Structure

This paper contains 46 sections, 38 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Preliminaries, assumptions, and notation
  3. Notation
  4. Polynomial coefficients for some rational functions
  5. Length functions and their entropy
  6. Length functions
  7. Algebraic entropy
  8. Intrinsic algebraic entropy
  9. Graded and filtered modules
  10. Graded modules
  11. Filtered modules
  12. Acceptable filterings and upward Artin-Rees Lemma
  13. Hilbert series for graded and filtered modules
  14. Filtered modules
  15. Hilbert polynomials for small modules
  16. ...and 31 more sections

Key Result

Proposition 2.1

Let $K$ be a ring of characteristic $0$, and $p(\bar{t}) \in K[\bar{t}]$. Let $\gamma_{1}, \dotsc, \gamma_{\ell} \in \mathbb{N}$. Define and expand $f$ as Then, there exists a polynomial $q(\bar{t}) \in K[\bar{t}]$ such that:

Theorems & Definitions (123)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Definition 3.5
  • Remark 3.6
  • Definition 4.1
  • ...and 113 more