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The syzygy theorem for Bézout rings

Maroua Gamanda, Henri Lombardi, Stefan Neuwirth, Ihsen Yengui

Abstract

We provide constructive versions of Hilbert's syzygy theorem for Z and Z/nZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict Bézout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.

The syzygy theorem for Bézout rings

Abstract

We provide constructive versions of Hilbert's syzygy theorem for Z and Z/nZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict Bézout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.

Paper Structure

This paper contains 14 sections, 19 theorems, 45 equations.

Table of Contents

  1. Gröbner bases for modules over a discrete ring
  2. The algorithms
  3. The context
  4. The division algorithm
  5. The S-polynomial algorithm
  6. Buchberger's algorithm
  7. The syzygy algorithm for terms
  8. Schreyer's syzygy algorithm
  9. The algorithms in the case of a valuation ring
  10. Termination of Buchberger's algorithm for a Bézout ring
  11. The syzygy theorem and Schreyer's algorithm for a valuation ring
  12. The syzygy theorem and Schreyer's algorithm for a Bézout ring
  13. The case of the integers
  14. The case of Z/NZ

Key Result

Lemma 4.1

Let $\mathbf{R}$ be a coherent strict Bézout ring with a divisibility test. Let $U$ be a submodule of $\mathbf{H}_{m}$ generated by a finite collection of terms $a_{\alpha}\underline{X}^{\alpha} e_{i_{\alpha}}$ with $\alpha \in A$. A term $b \underline{X}^{\beta} e_{r}$ lies in $U$ iff there is a no

Theorems & Definitions (44)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3: Monomial orders on finite-rank free $\mathbf{R}[\underline{X}]$-modules, see ALcoxlittleoshea05
  • Definition 1.4: Gröbner bases and Schreyer's monomial order
  • Remark 2.6
  • Remark 2.9
  • Example 3.3: S-polynomials over $\mathbf{R}=\mathbb{F}_{2}[Y]/\langle{Y^{r}}\rangle$, $r\geq 2$, a generalisation of Y5
  • Lemma 4.1: Term modules, see Y5
  • proof
  • Lemma 4.2
  • ...and 34 more