### Article

Kyungpook Mathematical Journal 2024; 64(2): 197-204

**Published online** June 30, 2024 https://doi.org/10.5666/KMJ.2024.64.2.197

Copyright © Kyungpook Mathematical Journal.

### Minimal Generators of Syzygy Modules Via Matrices

Haohao Wang* and Peter Oman

Department of Mathematics, Southeast Missouri State University, Cape Girardeau, MO 63701, USA

e-mail : hwang@semo.edu and poman@semo.edu

**Received**: April 20, 2023; **Revised**: November 18, 2023; **Accepted**: March 10, 2024

Let *R* = *x*] be a univariate polynomial ring over an algebraically closed field K of characteristic zero. Let *A* ∈ *M _{m,m}*(

*R*) be an m×m matrix over

*R*with non-zero determinate det(

*A*) ∈ R. In this paper, utilizing linear-algebraic techniques, we investigate the relationship between a basis for the syzygy module of

*f*

_{1}, . . . ,

*f*and a basis for the syzygy module of

_{m}*g*

_{1}, . . . ,

*g*, where [

_{m}*g*

_{1}, . . . ,

*g*] = [

_{m}*f*

_{1}, . . . ,

*f*]

_{m}*A*.

**Keywords**: Smith Normal Form, Cauchy-Binet Formula, Basis, Modules

### 1. Introduction

Systems of linear equations, vector spaces, and bases for vector spaces are widely studied in linear algebra. Modules are a generalization of vector spaces from linear algebra in which the “scalars" are allowed to be from an arbitrary ring, rather than a field. Many linear algebraic methods remain effective over principal ideal domains where the extended Euclidean algorithm may be utilized for computing with unimodular matrices. This paper is devoted to the some applications of linear algebra in studying modules over a univariate polynomial ring

Over the ring *R*, given

has a solution *b* is in the ideal generated by

is called a syzygy of *R* is essentially the syzygy problem, the goal is to find a generating set for the syzygy module.

A fundamental fact of linear algebra over a field that is a finitely generated vector space has a basis – the minimal generating (spanning) sets of a vector space are linearly independent and therefore form a basis. However, modules are more complicated than vector spaces; for instance, not all modules have a basis. It remains an interesting and an active research area to find a minimal generating set, or an upper bound for the size of a minimal generating set for different kinds of modules under various of conditions [2], [4], [6], and [8]. Hilbert's syzygy theorem [7] states that, if *M* is a finitely generated module over a multivariate polynomial ring *n*-th syzygy module of *M* is always a free module, i.e., a module that has a basis – a generating set consisting of linearly independent elements. In particular, over a univariate polynomial ring, Hilbert's syzygy theorem asserts that over a principal ideal ring, every submodule of a free module is itself free. Hence, it is an interesting problem to find the basis for syzygy modules.

This short paper investigates the minimal set of generators of a family of special modules. Our focus is centered on two finitely generated modules – the first module is Syz(**f**), the syzygy module of *m*-dimensional affine space. Similarly, *A* to the curve *R*.

A natural question to ask is “how to link a basis for *A*, and take advantage of Cauchy-Binet formula to relate a basis for

This paper is structured as the following. We begin in Section 2 with a brief review of results concerning a matrix factorization over PIDs, and the concepts of syzygy modules. We then answer the question that "how to link a basis for

### 2. A Brief Review

In this section, we provide a brief review of results concerning a matrix factorization over a PID, the concepts of syzygy modules, and the Hilbert-Burch theorem. First, we recall the Smith normal form over a PID and the Cauchy-Binet formula.

**Theorem 2.1.** (Smith Normal Form over PID [1, Theorem 3.1]) *Let R be a PID and let *

*where *

**Theorem 2.2.** (Cauchy-Binet formula [1, Theorem 2.34, Page 210]) *Suppose that A is an *

*where the sum extends over all subsets K of *

Next, we review a few concepts and results concerning syzygies, please refer [3, Chapters 4, 5] and [5, Chapter 20] for details. Recall that given a generating set *R*, a relation or first *syzygy* between the generators is a k-tuple

The set of syzygies form a module

Let *m*-1 elements – *μ*-basis, are expressed as the columns of a matrix *m*-1 of the matrix of *S*. That is, if *i*, and *m*-1 of the matrix of *S* without the *i*-th row, then the columns of *S* form a *μ*-basis if and only if

for some

Equation (2.2) is formulated from the following Hilbert–Burch theorem by applying *M*=*S* and

**Theorem 2.3.**(Hilbert–Burch Theorem [3, Proposition 2.6, Chater 6])

*Suppose that an ideal I in *

*Then there exists a non-zero element *

*M*obtained by deleting row

*i*. If

*I*has dimension

*n*-2, then we can take

*g*=1.

### 3. Main results

In this section, we investigate the relationship between a *μ*-basis for *μ*-basis for

We continue our notation – set

**Theorem 3.1.** *Let *

*Proof.* By Theorem 2.1,

where

Set *μ*-bases for

First, we claim that there exists a

To prove our claim, we note that the columns of *T*' form a *μ*-basis for

By the definition of *DT*' are elements of *S*', i.e.,

where

Now, we apply Cauchy-Binet formula in Theorem 2.2 to both sides of Equation (3.3). First, we compute all *i*-th row for

The last equality holds since the columns of *S*' form a *μ*-basis for

Again, applying Equation (2.1), we compute all *i*-th row for

where the last equality is due to Equation (2.2). It is easy to observe that the diagonal matrix *D* in Equation (3.1) has the property that

Thus,

In addition, the equality

yields that

Therefore, the Equation (3.3) yields that for each

and the Equations (3.4) and (3.5) show that

where

Hence, we have proved that there exists a

Now we are ready to prove the statement of the theorem. Note that on one hand,

that is, columns of *μ*-basis for

that is, columns of *μ*-basis for

where *μ*-bases are unique up-to isomorphism, without loss of generality, we may set *T*'=*VT*. Applying a similar argument to *S* and *S*'. We conclude that *μ*-bases for Syz(**g**) and *T* and *S* respectively where

Thus, replacing *T*' and *S*' by *VT* and

Therefore, we conclude that there exists a

Below, we will provide a simple example to illustrate Theorem 3.1.

**Example 3.1.** Consider *μ*-basis of *S*, with an appropriate sign, give

Let

and

A *μ*-basis for *T*, with an appropriate sign, give

We have that

### 4. Conclusions

Applying linear-algebraic techniques, we obtained a relationship on the minimal generators *S* and *T* are *μ*-bases for

We took advantage the Smith normal form of the matrix *A* over

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