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Componentwise linear syzygies and good almost regular sequences

Satoru Isogawa

TL;DR

This work develops a framework of $\gamma$-regular sequences to characterize when graded modules over a polynomial ring have componentwise linear syzygies. By introducing $\gamma$-depth and its stabilized variant $\hat{\gamma}$-depth, the paper proves that a finitely generated graded module $M$ over $R=K[x_1,\dots,x_n]$ has a componentwise linear first syzygy if and only if $\gamma\text{-depth}_R M=n$, using successive reductions along $\gamma$-regular elements. It provides a suite of criteria and equivalences for single elements and sequences, with detailed behavior of the first syzygy under reductions and a splitting formula for Betti numbers. The results extend the Harima–Watanabe characterization via the $\mathfrak m$-full property, delivering structural descriptions for finite-dimensional modules and a two-variable polynomial ring case that clarifies the classification of linear syzygies and Poincaré series. Overall, the paper advances a robust, dimension-reducing approach to componentwise linearity in the graded setting, with potential implications for Koszulness and syzygetic behavior in broader contexts.

Abstract

In this paper, we introduce the notion of $γ$-regular sequences to characterize the property that graded modules have componentwise linear syzygies. This extends Harima and Watanabe's characterization of componentwise linear ideals in terms of $\mathfrak{m}$-full property.

Componentwise linear syzygies and good almost regular sequences

TL;DR

This work develops a framework of -regular sequences to characterize when graded modules over a polynomial ring have componentwise linear syzygies. By introducing -depth and its stabilized variant -depth, the paper proves that a finitely generated graded module over has a componentwise linear first syzygy if and only if , using successive reductions along -regular elements. It provides a suite of criteria and equivalences for single elements and sequences, with detailed behavior of the first syzygy under reductions and a splitting formula for Betti numbers. The results extend the Harima–Watanabe characterization via the -full property, delivering structural descriptions for finite-dimensional modules and a two-variable polynomial ring case that clarifies the classification of linear syzygies and Poincaré series. Overall, the paper advances a robust, dimension-reducing approach to componentwise linearity in the graded setting, with potential implications for Koszulness and syzygetic behavior in broader contexts.

Abstract

In this paper, we introduce the notion of -regular sequences to characterize the property that graded modules have componentwise linear syzygies. This extends Harima and Watanabe's characterization of componentwise linear ideals in terms of -full property.

Paper Structure

This paper contains 11 sections, 33 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Componentwise linear modules
  3. $\gamma$-regular elements
  4. $\gamma$-regular sequences
  5. Criteria for a $\gamma$-regular element
  6. Criteria for a $\gamma$-regular sequence
  7. $\hat{\gamma}$-regular elements and $\hat{\gamma}$-regular sequences
  8. Modules with a first syzygy generated in a single degree
  9. $\hat{\gamma}\text{-sequence}$ revisited
  10. Finite dimensional modules with linear syzygies
  11. Modules with linear syzygies over a polynomial ring in two variables

Key Result

Theorem 1

Let $K$ be an infinite field and $M$ be a finitely generated graded module over the polynomial ring $R=K\lbrack x_1, \cdots, x_n \rbrack$. Then the following are equivalent:

Theorems & Definitions (90)

  • Theorem
  • Theorem 1.1: Harima and Watanabe
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 3.1
  • Remark 3.3
  • Definition 3.4
  • Remark 3.6
  • Lemma 3.7
  • ...and 80 more