Posted on Categories:Commutative Algebra, 交换代数, 数学代写

# 数学代写|交换代数代写Commutative Algebra代考|Computing Resolutions and the Syzygy Theorem

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|交换代数代写Commutative Algebra代考|Computing Resolutions and the Syzygy Theorem

Let $K$ be a field and $>$ a monomial ordering on $K[x]^\tau$. Again $R$ denotes the localization of $K[x]$ with respect to $S_{>}$.

We shall give a method, using standard bases, to compute syzygies and, more generally, free resolutions of finitely generated $R$-modules. Syzygies and free resolutions are very important objects and basic ingredients for many constructions in homological algebra and algebraic geometry. On the other hand, the use of syzygies gives a very elegant way to prove Buchberger’s criterion for standard bases. Moreover, a close inspection of the syzygies of the generators of an ideal allows detection of useless pairs during the computation of a standard basis.

In the following definition $R$ can be an arbitrary ring.
Definition 2.5.1. A syzygy or relation between $k$ elements $f_1, \ldots, f_k$ of an $R$-module $M$ is a $k$-tuple $\left(g_1, \ldots, g_k\right) \in R^k$ satisfying
$$\sum_{i=1}^k g_i f_i=0$$
The set of all syzygies between $f_1, \ldots, f_k$ is a submodule of $R^k$. Indeed, it is the kernel of the ring homomorphism
$$\varphi: F_1:=\bigoplus_{i=1}^k R \varepsilon_i \longrightarrow M, \quad \varepsilon_i \longmapsto f_i,$$
where $\left{\varepsilon_1, \ldots, \varepsilon_k\right}$ denotes the canonical basis of $R^k . \varphi$ surjects onto the $R-\operatorname{module} I:=\left\langle f_1, \ldots, f_k\right\rangle_R$ and
$$\operatorname{syz}(I):=\operatorname{syz}\left(f_1, \ldots, f_k\right):=\operatorname{Ker}(\varphi)$$
is called the module of syzygies of $I$ with respect to the generators $f_1, \ldots, f_k{ }^8$

## 数学代写|交换代数代写Commutative Algebra代考|Modules over Principal Ideal Domains

In this section we shall study the structure of finitely generated modules over principal ideal domains. It will be proved that they can be decomposed in a unique way into a direct sum of cyclic modules with special properties. Examples are given for the case of a univariate polynomial ring over a field. We show how this decomposition can be computed by using standard bases (actually, we need only interreduction).

Theorem 2.6.1. Let $R$ be a principal ideal domain and $M$ a finitely generated $R$-module, then $M$ is a direct sum of cyclic modules.

Proof. Let $R^m \rightarrow R^n \rightarrow M \rightarrow 0$ be a presentation of $M$ given by the ma$\operatorname{trix} A=\left(a_{i j}\right)$ with respect to the bases $B=\left{e_1, \ldots, e_n\right}, B^{\prime}=\left{f_1, \ldots, f_m\right}$ of $R^n, R^m$, respectively. If $A$ is the zero-matrix, then $M \cong R^n$, and we are done. Otherwise, we may assume that $a_{11} \neq 0$. We shall show that, for a suitable choice of the bases, the presentation matrix has diagonal form, that is, $a_{i j}=0$ if $i \neq j$. For some $k>1$ with $a_{k 1} \neq 0$, let $h$ be a generator of the ideal $\left\langle a_{11}, a_{k 1}\right\rangle$, and let $a, b, c, d \in R$ be such that $h=a a_{11}+b a_{k 1}, a_{11}=c h$, $a_{k 1}=d h$ (we choose $a:=1, b:=0, c:=1$ if $\left\langle a_{11}\right\rangle=\left\langle a_{11}, a_{k 1}\right\rangle$ ). Now we change the basis $B$ to $\bar{B}=\left{c e_1+d e_k, e_2, \ldots, e_{k-1},-b e_1+a e_k, e_{k+1}, \ldots, e_n\right}$. $\bar{B}$ is a basis because $\operatorname{det}\left(\begin{array}{cc}c & -b \ d & a\end{array}\right)=1$. Let $\bar{A}=\left(\bar{a}{i j}\right)$ be the presentation matrix with respect to this basis, then $\bar{a}{11}=h$ and $\bar{a}{k 1}=0$, while $\bar{a}{i 1}=a_{i 1}$ for $i \neq 1, k$. Note that the first row of $A$ and $\bar{A}$ are equal if and only if $\left\langle a_{11}\right\rangle=\left\langle a_{11}, a_{k 1}\right\rangle$. Doing this with every $k>1$, we may assume that $a_{k 1}=0$ for $k=2, \ldots, n$.

Now, applying the same procedure to the transposed matrix ${ }^t A$ (which corresponds to base changes in $B^{\prime}$ ), we obtain a matrix ${ }^t A_1$,
$$A_1=\left(\begin{array}{cccc} a_{11}^{(1)} & 0 & \ldots & 0 \ a_{21}^{(1)} & a_{22}^{(1)} & \ldots & a_{2 m}^{(1)} \ \vdots & & & \vdots \ a_{n 1}^{(1)} & a_{n 2}^{(1)} & \ldots & a_{n m}^{(1)} \end{array}\right),$$
with the property: $\left\langle a_{11}\right\rangle \subset\left\langle a_{11}^{(1)}\right\rangle$ and $a_{21}^{(1)}=\cdots=a_{n 1}^{(1)}=0$, if $\left\langle a_{11}\right\rangle=\left\langle a_{11}^{(1)}\right\rangle$.

## 数学代写|交换代数代写Commutative Algebra代考|Computing Resolutions and the Syzygy Theorem

2.5.1.定义$R$ -模块$M$的$k$元素$f_1, \ldots, f_k$之间的聚合或关系是一个$k$ -元组$\left(g_1, \ldots, g_k\right) \in R^k$
$$\sum_{i=1}^k g_i f_i=0$$
$f_1, \ldots, f_k$之间所有协同的集合是$R^k$的一个子模块。事实上，它是环同态的核
$$\varphi: F_1:=\bigoplus_{i=1}^k R \varepsilon_i \longrightarrow M, \quad \varepsilon_i \longmapsto f_i,$$

$$\operatorname{syz}(I):=\operatorname{syz}\left(f_1, \ldots, f_k\right):=\operatorname{Ker}(\varphi)$$

## 数学代写|交换代数代写Commutative Algebra代考|Modules over Principal Ideal Domains

$$A_1=\left(\begin{array}{cccc} a_{11}^{(1)} & 0 & \ldots & 0 \ a_{21}^{(1)} & a_{22}^{(1)} & \ldots & a_{2 m}^{(1)} \ \vdots & & & \vdots \ a_{n 1}^{(1)} & a_{n 2}^{(1)} & \ldots & a_{n m}^{(1)} \end{array}\right),$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。