The Bourbaki degree of the syzygy module of 2 $\times$ 4 matrices

Abstract: We introduce and study the Bourbaki degree as a numerical invariant for (2 \times 4) matrices $Θ$ of homogeneous polynomials over a polynomial ring (R = k[x_1, \dots, x_n]). This invariant, defined via a Bourbaki sequence for the syzygy module (\operatorname{Syz}(Θ)), generalizes previous constructions for plane curves and Jacobian matrices. Our main result is an explicit formula expressing the Bourbaki degree in terms of the degrees of the rows, the initial degree of a syzygy, and the first two Hilbert coefficients of the cokernel module (\mathcal{Q} = \operatorname{coker}(Θ)). We apply this framework to two important cases. First, matrices with constant first row, which are determined by a three-equigenerated ideal (J = (f_1, f_2, f_3)), where we show the Bourbaki degree measures how far (J) is from being a perfect ideal, and we completely characterize its smaller and larger values. Second, for a linear matrix, we use the Kronecker--Weierstrass classification to determine all possible Bourbaki degrees and homological types. This classification reveals the existence of a linear matrix with Bourbaki degree equal to 2, a value that does not occur for Jacobian matrices. Finally, in the geometric context of (\mathbb{P}^3), we provide a sufficient condition for (\operatorname{Syz}(Θ)) to define a codimension one distribution and obtain bounds on the Bourbaki degree when the initial degree is small.

• The paper introduces the Bourbaki degree as a new numerical invariant capturing the deviation from freeness in 2×4 syzygy modules.
• It provides an explicit formula that connects the invariant with the degrees of matrix rows, the initial degree of syzygies, and Hilbert coefficients, validated over various matrix types.
• Special cases, including matrices with a constant first row and linear matrices, are analyzed to reveal geometric associations with codimension one distributions in projective spaces.

Bourbaki Degree as a Numerical Invariant for Syzygy Modules of $2 \times 4$ Matrices

Introduction and Motivation

This paper introduces the Bourbaki degree as a new numerical invariant for $2 \times 4$ matrices $\Theta$ of homogeneous polynomials over a polynomial ring $R = k[x_1, \dots, x_n]$, with $k$ an infinite field. The invariant is constructed via Bourbaki sequences associated to the syzygy module $\operatorname{Syz}(\Theta)$, generalizing previous approaches for Jacobian matrices and plane curves. The work establishes an explicit formula for the Bourbaki degree, demonstrating its connection to the degrees of the matrix rows, the initial degree of syzygies, and Hilbert coefficients of the cokernel $\mathcal{Q} = \operatorname{coker}(\Theta)$.

Major applications are developed for matrices with special structure: those with a constant first row, linking the Bourbaki degree to equigenerated ideals and their departure from perfection; and linear matrices, classified via the Kronecker–Weierstrass normal form. The geometric implications are also explored, particularly in $\mathbb{P}^3$, where the syzygy module relates to codimension one distributions.

Homological and Algebraic Framework

Let $\Theta$ be a $2 \times 4$ matrix of rank $2 \times 4$0, with rows composed of homogeneous polynomials of degrees $2 \times 4$1 and $2 \times 4$2, respectively. The primary object is the syzygy module $2 \times 4$3, a reflexive $2 \times 4$4-module of rank $2 \times 4$5. The paper investigates the Buchsbaum–Rim complex as the graded free resolution for $2 \times 4$6 when the ideal of $2 \times 4$7 minors $2 \times 4$8 has maximal grade, and delineates the cases in which the syzygy module is free or locally free.

A crucial parameter is the initial degree $2 \times 4$9, with constraints $\Theta$0, arising from interaction between row-wise syzygy modules. The Hilbert polynomial of $\Theta$1, with coefficients $\Theta$2 and $\Theta$3, governs much of the subsequent numerical analysis.

Definition and Explicit Formula for the Bourbaki Degree

The central construction starts with a minimal homogeneous syzygy $\Theta$4 of degree $\Theta$5, yielding an injective map $\Theta$6. The quotient is a rank one torsion-free module, isomorphic to an ideal $\Theta$7 up to degree shift. The Bourbaki degree, $\Theta$8, is defined as $\Theta$9, encapsulating how far $R = k[x_1, \dots, x_n]$0 is from being free.

The main theorem delivers the explicit formula: $R = k[x_1, \dots, x_n]$1 where $R = k[x_1, \dots, x_n]$2, and $R = k[x_1, \dots, x_n]$3, $R = k[x_1, \dots, x_n]$4.

In the maximal grade case ($R = k[x_1, \dots, x_n]$5), $R = k[x_1, \dots, x_n]$6 and the Buchsbaum–Rim complex resolves $R = k[x_1, \dots, x_n]$7.

Specializations: Three-Equigenerated Ideals and Linear Matrices

Three-Equigenerated Ideals

For matrices with a constant first row, the associated ideal $R = k[x_1, \dots, x_n]$8 implies $R = k[x_1, \dots, x_n]$9 defines a Bourbaki degree for $k$0, denoted $k$1. The results (Main Theorem~\ref{BourJ}) provide a complete characterization of extremal values:

• $k$2 iff $k$3 is perfect.
• $k$4 iff $k$5 is a complete intersection of two linear forms.
• $k$6 iff $k$7 is the intersection of two linear primes, a linear primary ideal of multiplicity two, or a complete intersection of type $k$8.
• $k$9 iff $\operatorname{Syz}(\Theta)$0 and $\operatorname{Syz}(\Theta)$1.
• $\operatorname{Syz}(\Theta)$2 iff $\operatorname{Syz}(\Theta)$3 is a complete intersection.

A key assertion is that $\operatorname{Syz}(\Theta)$4, for locally free $\operatorname{Syz}(\Theta)$5, satisfies $\operatorname{Syz}(\Theta)$6 and $\operatorname{Syz}(\Theta)$7.

Linear Matrices

A rigorous analysis leveraging the Kronecker–Weierstrass classification yields that each canonical matrix type realizes a distinct Bourbaki degree and homological profile:

• Most types are either free, nearly free ($\operatorname{Syz}(\Theta)$8), or Buchsbaum–Rim ($\operatorname{Syz}(\Theta)$9).
• Remarkably, the matrix $\mathcal{Q} = \operatorname{coker}(\Theta)$0 produces $\mathcal{Q} = \operatorname{coker}(\Theta)$1, a value not realized for Jacobian matrices, contradicting previous classifications for pencils of quadrics [Faenzi2025].
• Minimal free resolutions for $\mathcal{Q} = \operatorname{coker}(\Theta)$2 are systematically determined by the block type, with explicit formulas for the shifts.

Geometric Context: Distributions and Logarithmic Sheaves

For $\mathcal{Q} = \operatorname{coker}(\Theta)$3, the paper connects syzygy modules to codimension one distributions on $\mathcal{Q} = \operatorname{coker}(\Theta)$4, via compositions with the Euler vector. If the associated polynomials form a regular sequence, $\mathcal{Q} = \operatorname{coker}(\Theta)$5 is interpreted as the tangent sheaf of such a distribution. For Jacobian matrices of regular sequences, the induced distribution is integrable.

Sharp bounds on $\mathcal{Q} = \operatorname{coker}(\Theta)$6 are obtained in low initial degree cases, exploiting the geometry of foliations and curves. Nearly free matrices cannot induce locally free syzygy modules, extending established facts for Jacobian matrices.

Structural and Numerical Implications

Across all cases, the Bourbaki degree provides a discrete measurement of structural deviation from freeness or perfection:

• For ideals or modules, it captures codimension two defects and links homological properties to geometric singularities.
• The gaps and extremal values show that certain numerical invariants are unattainable, reflecting deep constraints in the algebraic structure.
• Explicit bounds on the Bourbaki degree often encode geometric restrictions, such as non-saturation, multiplicity, or dimension limitations.

In the context of projective geometry, syzygy modules with prescribed Bourbaki degree inform classification schemes for distributions, null-correlation bundles, and singular foliations.

Future Directions

The formalism established paves the way toward broader applications:

• Extending Bourbaki degree computations to mixed degree cases, such as $\mathcal{Q} = \operatorname{coker}(\Theta)$7, relevant for complete intersection curves in projective space.
• Integrating the invariant with classification theory for logarithmic sheaves, higher codimension arrangements, and singularities.
• Studying the relationship between the Bourbaki degree and geometric moduli spaces or stability conditions.

There is also a compelling computational angle, suggesting algorithmic realization in computer algebra systems for explicit calculations in more complex settings.

Conclusion

The paper systematically develops and analyzes the Bourbaki degree for syzygy modules of $\mathcal{Q} = \operatorname{coker}(\Theta)$8 matrices, providing a robust formula encapsulating crucial numerical and homological invariants. The results bridge commutative algebra, homological algebra, and algebraic geometry, enabling precise classification of ideals, syzygy modules, and their associated geometric objects, and offering practical bounds and a structural lens for interpreting algebraic and geometric complexity (2604.04252).