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## 数学代写|交换代数代写Commutative Algebra代考|Basic definitions

Suppose $(M,+)$ is an abelian group. For any $m \in M$ and any integer $n$, one can make sense of $n \bullet m$. If $n$ is a positive integer, this means $m+\cdots+m$ ( $n$ times); if $n=0$ it means 0 , and if $n$ is negative, then $n \bullet m=-(-n) \bullet m$. Thus we have defined a function $\bullet: \mathbb{Z} \times M \rightarrow M$ which enjoys the following properties: for all $n, n_1, n_2 \in \mathbb{Z}, m, m_1, m_2 \in M$, we have
(ZMOD1) $1 \bullet m=m$.
(ZMOD2) $n \bullet\left(m_1+m_2\right)=n \bullet m_1+n \bullet m_2$.
(ZMOD3) $\left(n_1+n_2\right) \bullet m=n_1 \bullet m+n_2 \bullet m$.
(ZMOD4) $\left(n_1 n_2\right) \bullet m=n_1 \bullet\left(n_2 \bullet m\right)$
It should be clear that this is some kind of ring-theoretic analogue of a group action on a set. In fact, consider the slightly more general construction of a monoid $(M, \cdot)$ acting on a set $S$ : that is, for all $n_1, n_2 \in M$ and $s \in S$, we require $1 \bullet s=s$ and $\left(n_1 n_2\right) \bullet s=n_1 \bullet\left(n_2 \bullet s\right)$.

For a group action $G$ on $S$, each function $g \bullet: S \rightarrow S$ is a bijection. For monoidal actions, this need not hold for all elements: e.g. taking the natural multiplication action of $M=(\mathbb{Z}, \cdot)$ on $S=\mathbb{Z}$, we find that $0 \bullet: \mathbb{Z} \rightarrow{0}$ is neither injective nor surjective, $\pm 1 \bullet: \mathbb{Z} \rightarrow \mathbb{Z}$ is bijective, and for $|n|>1, n \bullet: \mathbb{Z} \rightarrow \mathbb{Z}$ is injective but not surjective.

## 数学代写|交换代数代写Commutative Algebra代考|Finitely presented modules

One of the major differences between abelian groups and nonabelian groups is that a subgroup $N$ of a finitely generated abelian group $M$ remains finitely generated, and indeed, the minimal number of generators of the subgroup $N$ cannot exceed the minimal number of generators of $M$, whereas this is not true for nonabelian groups: e.g. the free group of rank 2 has as subgroups free groups of every rank $0 \leq r \leq \aleph_0$. (For instance, the commutator subgroup is not finitely generated.)
Since an abelian group is a $\mathbb{Z}$-module and every $R$-module has an underlying abelian group structure, one might well expect the situation for $R$-modules to be similar to that of abelian groups. We will see later that this is true in many but not all cases: an $R$-module is called Noetherian if all of its submodules are finitely generated. Certainly a Noetherian module is itself finitely generated. The basic fact here which we will prove in $\S 8.7$ – is a partial converse: if the ring $R$ is Noetherian, any finitely generated $R$-module is Noetherian. Note that we can already see that the Noetherianity of $R$ is necessary: if $R$ is not Noetherian, then by definition there exists an ideal $I$ of $R$ which is not finitely generated, and this is nothing else than a non-finitely generated $R$-submodule of $R$ (which is itself generated by the single element 1.) Thus the aforementioned fact about subgroups of finitely generated abelian groups being finitely generated holds because $\mathbb{Z}$ is a Noetherian ring.
When $R$ is not Noetherian, it becomes necessary to impose stronger conditions than finite generation on modules. One such condition indeed comes from group theory: recall that a group $G$ is finitely presented if it is isomorphic to the quotient of a finitely generated free group $F$ by the least normal subgroup $N$ generated by a finite subset $x_1, \ldots, x_m$ of $F$.

## 数学代写|交换代数代写Commutative Algebra代考|Basic definitions

(zmod1) $1 \bullet m=m$．
(zmod2) $n \bullet\left(m_1+m_2\right)=n \bullet m_1+n \bullet m_2$．
(zmod3) $\left(n_1+n_2\right) \bullet m=n_1 \bullet m+n_2 \bullet m$．
(zmod4) $\left(n_1 n_2\right) \bullet m=n_1 \bullet\left(n_2 \bullet m\right)$

## 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。

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## 数学代写|交换代数代写Commutative Algebra代考|The basic formalism

Let $(X, \leq)$ be a partially ordered set. We denote by $X^{\vee}$ the order dual of $X$ : it has the same underlying set as $X$ but the inverse order relation: $x \preceq y \Longleftrightarrow y \leq x$.
Let $(X, \leq)$ and $(Y, \leq)$ be partially ordered sets. A map $f: X \rightarrow Y$ is isotone (or order-preserving) if for all $x_1, x_2 \in X, x_1 \leq x_2 \Longrightarrow f\left(x_1\right) \leq f\left(x_2\right) ; f$ is antitone (or order-reversing) if for all $x_1, x_2 \in X, x_1 \leq x_2 \Longrightarrow f\left(x_1\right) \geq f\left(x_2\right)$.
Exercise 2.1. Let $X, Y, Z$ be partially ordered sets, and let $f: X \rightarrow Y, g$ : $Y \rightarrow Z$ be functions. Show:
a) If $f$ and $g$ are isotone, then $g \circ f$ is isotone.
b) If $f$ and $g$ are antitone, then $g \circ f$ is isotone.
c) If one of $f$ and $g$ is isotone and the other is antitone, then $g \circ f$ is antitone.
Let $(X, \leq)$ and $(Y, \leq)$ be partially ordered sets. An antitone Galois connection between $\mathbf{X}$ and $\mathbf{Y}$ is a pair of maps $\Phi: X \rightarrow Y$ and $\Psi: Y \rightarrow X$ such that:
(GC1) $\Phi$ and $\Psi$ are both antitone maps, and
(GC2) For all $x \in X$ and all $y \in Y, x \leq \Psi(y) \Longleftrightarrow y \leq \Phi(x)$.
There is a pleasant symmetry in the definition: if $(\Phi, \Psi)$ is a Galois connection between $X$ and $Y$, then $(\Psi, \Phi)$ is a Galois connection between $Y$ and $X$.

If $(X, \leq)$ is a partially ordered set, then a mapping $f: X \rightarrow X$ is called a closure operator if it satisfies all of the following properties:
(C1) For all $x \in X, x \leq f(x)$.
(C2) For all $x_1, x_2 \in X, x_1 \leq x_2 \Longrightarrow f\left(x_1\right) \leq f\left(x_2\right)$.
(C3) For all $x \in X, f(f(x))=f(x)$.

## 数学代写|交换代数代写Commutative Algebra代考|Lattice properties

Recall that a partially ordered set $X$ is a lattice if for all $x_1, x_2 \in X$, there is a greatest lower bound $x_1 \wedge x_2$ and a least upper bound $x_1 \vee x_2$. A partially ordered set is a complete lattice if for every subset $A$ of $X$, the greatest lower bound $\wedge A$ and the least upper bound $\bigvee A$ both exist.
Lemma 2.4. Let $(X, Y, \Phi, \Psi)$ be a Galois connection.
a) If $X$ and $Y$ are both lattices, then for all $x_1, x_2 \in X$,
\begin{aligned} & \Phi\left(x_1 \wedge x_2\right)=\Phi\left(x_1\right) \vee \Phi\left(x_2\right), \ & \Phi\left(x_2 \vee x_2\right)=\Phi\left(x_1\right) \wedge \Phi\left(x_2\right) . \end{aligned}
b) If $X$ and $Y$ are both complete lattices, then for all subsets $A \subset X$,
\begin{aligned} & \Phi(\bigwedge A)=\bigvee \Phi(A), \ & \Phi(\bigvee A)=\bigwedge \Phi(A) . \end{aligned}
Exercise 2.2. Prove Lemma 2.4.
Complete lattices also intervene in this subject in the following way.
Proposition 2.5. Let $A$ be a set and let $X=\left(2^A, \subset\right)$ be the power set of $A$, partially ordered by inclusion. Let $c: X \rightarrow X$ be a closure operator. Then the collection $c(X)$ of closed subsets of $A$ forms a complete lattice, with $\wedge S=\bigcap_{B \in S} B$ and $\bigvee S=c\left(\bigcup_{B \in S} B\right)$

## 数学代写|交换代数代写Commutative Algebra代考|The basic formalism

a)如果 $f$ 和 $g$ 是等音的吗 $g \circ f$ 是等音的。
b)如果 $f$ 和 $g$ 是反调吗 $g \circ f$ 是等音的。
c)如果其中之一 $f$ 和 $g$ 一个是等音，另一个是反音 $g \circ f$ 是反调。

(gc1) $\Phi$ 和 $\Psi$ 两者都是反调地图吗
(GC2)对所有人 $x \in X$ 等等 $y \in Y, x \leq \Psi(y) \Longleftrightarrow y \leq \Phi(x)$．

(C1)所有人$x \in X, x \leq f(x)$。
(C2)所有人$x_1, x_2 \in X, x_1 \leq x_2 \Longrightarrow f\left(x_1\right) \leq f\left(x_2\right)$。
(C3)对所有人$x \in X, f(f(x))=f(x)$。

## 数学代写|交换代数代写Commutative Algebra代考|Lattice properties

a)如果$X$和$Y$都是格，则对于所有$x_1, x_2 \in X$，
\begin{aligned} & \Phi\left(x_1 \wedge x_2\right)=\Phi\left(x_1\right) \vee \Phi\left(x_2\right), \ & \Phi\left(x_2 \vee x_2\right)=\Phi\left(x_1\right) \wedge \Phi\left(x_2\right) . \end{aligned}
b)如果$X$和$Y$都是完全格，则对于所有子集$A \subset X$，
\begin{aligned} & \Phi(\bigwedge A)=\bigvee \Phi(A), \ & \Phi(\bigvee A)=\bigwedge \Phi(A) . \end{aligned}

## 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。

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

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## 数学代写|交换代数代写Commutative Algebra代考|The Category of Finitely Presented Modules

The category of finitely presented modules over A can be constructed from the category of free modules of finite rank over $\mathbf{A}$ by a purely categorical procedure.

1. A finitely presented module $M$ is described by a triplet
$$\left(\mathrm{K}_M, \mathrm{G}_M, \mathrm{~A}_M\right),$$
where $\mathrm{A}_M$ is a linear map between the free modules of finite ranks $\mathrm{K}_M$ and $\mathrm{G}_M$. We have $M \simeq$ Coker $\mathrm{A}_M$ and $\pi_M: \mathrm{G}_M \rightarrow M$ is the surjective linear map with kernel $\operatorname{Im} \mathrm{A}_M$. The matrix of the linear map $\mathrm{A}_M$ is a presentation matrix of $M$.
2. A linear map $\varphi$ of the module $M$ (described by $\left(\mathrm{K}M, \mathrm{G}_M, \mathrm{~A}_M\right)$ ) to the module $N$ (described by $\left(\mathrm{K}_N, \mathrm{G}_N, \mathrm{~A}_N\right)$ ) is described by two linear maps $\mathrm{K}{\varphi}: \mathrm{K}M \rightarrow \mathrm{K}_N$ and $\mathrm{G}{\varphi}: \mathrm{G}M \rightarrow \mathrm{G}_N$ subject to the commutation relation $\mathrm{G}{\varphi} \circ \mathrm{A}M=\mathrm{A}_N \circ \mathrm{K}{\varphi}$.
1. The sum of two linear maps $\varphi$ and $\psi$ of $M$ to $N$ represented by $\left(\mathrm{K}{\varphi}, \mathrm{G}{\varphi}\right)$ and $\left(\mathrm{K}\psi, \mathrm{G}\psi\right)$ is represented by $\left(\mathrm{K}{\varphi}+\mathrm{K}\psi, \mathrm{G}{\varphi}+\mathrm{G}\psi\right)$.
The linear map $a \varphi$ is represented by $\left(a \mathrm{~K}{\varphi}, a \mathrm{G}{\varphi}\right)$.
2. To represent the composite of two linear maps, we compose their representations.
3. Finally, the linear map $\varphi$ of $M$ to $N$ represented by $\left(\mathrm{K}{\varphi}, \mathrm{G}{\varphi}\right)$ is null if and only if there exists a $Z_{\varphi}: \mathrm{G}M \rightarrow \mathrm{K}_N$ satisfying $\mathrm{A}_N \circ Z{\varphi}=\mathrm{G}_{\varphi}$.

## 数学代写|交换代数代写Commutative Algebra代考|Stability Properties

4.1 Proposition Let $N_1$ and $N_2$ be two finitely generated $\mathbf{A}$-submodules of an $\mathbf{A}$ module $M$. If $N_1+N_2$ is finitely presented, then $N_1 \cap N_2$ is finitely generated.
D We can follow almost word for word the proof of item $I$ of Theorem II-3.4 (necessary condition).
4.2 Proposition Let $N$ be an A-submodule of $M$ and $P=M / N$.

1. If $M$ is finitely presented and $N$ finitely generated, then $P$ is finitely presented.
2. If $M$ is finitely generated and $P$ finitely presented, then $N$ is finitely generated.
3. If $P$ and $N$ are finitely presented, then $M$ is finitely presented. More precisely, if $A$ and $B$ are presentation matrices for $N$ and $P$, we have a presentation matrix $D=$\begin{tabular}{|l|l|} \hline$A$ & $C$ \ \hline 0 & $B$ \ \hline \end{tabular}

D 1. We can suppose that $M=\mathbf{A}^p / F$ with $F$ finitely generated. If $N$ is finitely generated, it is of the form $N=\left(F^{\prime}+F\right) / F$ where $F^{\prime}$ is finitely generated, so $P \simeq \mathbf{A}^p /\left(F+F^{\prime}\right)$.

We write $M=\mathbf{A}^p / F$ and $N=\left(F^{\prime}+F\right) / F$. We have $P \simeq \mathbf{A}^p /\left(F^{\prime}+F\right)$, so $F^{\prime}+F($ and also $N)$ is finitely generated (Sect. 1).

## 数学代写|交换代数代写Commutative Algebra代考|The Category of Finitely Presented Modules

A上有限呈现模的范畴可以由$\mathbf{A}$上有限秩的自由模的范畴用纯范畴的方法构造。

$$\left(\mathrm{K}_M, \mathrm{G}_M, \mathrm{~A}_M\right),$$

## 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。

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

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## 数学代写|交换代数代写Commutative Algebra代考|Algebraic Identities

An algebraic identity is an equality between two elements of $\mathbb{Z}\left[X_1, \ldots, X_n\right]$ defined differently. It gets automatically transferred into every commutative ring by means of the previous corollary.

Since the ring $\mathbb{Z}\left[X_1, \ldots, X_n\right]$ has particular properties, it happens that some algebraic identities are easier to prove in $\mathbb{Z}\left[X_1, \ldots, X_n\right]$ than in “an arbitrary ring $\mathbf{B}$.” Consequently, if the structure of a theorem reduces to a family of algebraic identities, which is very frequent in commutative algebra, it is often in our interest to use a ring of polynomials with coefficients in $\mathbb{Z}$ by taking as its indeterminates the relevant elements in the statement of the theorem.

The properties of the rings $\mathbb{Z}[\underline{X}]$ which may prove useful are numerous. The first is that it is an integral ring. So it is a subring of its quotient field $\mathbb{Q}\left(X_1, \ldots, X_n\right)$ which offers all the facilities of discrete fields.

The second is that it is an infinite and integral ring. Consequently, “all bothersome but rare cases can be ignored.” A case is rare when it corresponds to the annihilation of a polynomial $Q$ that evaluates to zero everywhere. It suffices to check the equality corresponding to the algebraic identity when it is evaluated at the points of $\mathbb{Z}^n$ which do not annihilate $Q$. Indeed, if the algebraic identity we need to prove is $P=0$, we get that the polynomial $P Q$ defines the function over $\mathbb{Z}^n$ that evaluates to zero everywhere, this implies that $P Q=0$ and thus $P=0$ since $Q \neq 0$ and $\mathbb{Z}[\underline{X}]$ is integral. This is sometimes called the “extension principle for algebraic identities.”
Other remarkable properties of $\mathbb{Z}[\underline{X}]$ could sometimes be used, like the fact that it is a unique factorization domain (UFD) as well as being a strongly discrete coherent Noetherian ring of finite Krull dimension.

## 数学代写|交换代数代写Commutative Algebra代考|Weights, Homogeneous Polynomials

We say that we have defined a weight on a polynomial algebra $\mathbf{A}\left[X_1, \ldots, X_k\right]$ when we attribute to each indeterminate $X_i$ a weight $w\left(X_i\right) \in \mathbb{N}$. We then define the weight of the monomial $\underline{X} \underline{\underline{m}}=X_1^{m_1} \cdots X_k^{m_k}$ as
$$w\left(\underline{X}^{\underline{m}}\right)=\sum_i m_i w\left(X_i\right)$$
so that $w\left(\underline{X}^{\underline{m}}+\underline{m^{\prime}}\right)=w\left(\underline{X}^{\underline{m}}\right)+w\left(\underline{X}^{m^{\prime}}\right)$. The degree of a polynomial $P$ for this weight, generally denoted by $w(P)$, is the greatest of the weights of the monomials appearing with a nonzero coefficient. This is only well-defined if we have a test of equality to 0 in $\mathbf{A}$ at our disposal. In the opposite case we simply define the statement ” $w(P) \leqslant r . “$

A polynomial is said to be homogeneous (for a weight $w$ ) if all of its monomials have the same weight.

When we have an algebraic identity and a weight available, each homogeneous component of the algebraic identity provides a particular algebraic identity.

We can also define weights with values in some monoids with a more complicated order than $(\mathbb{N}, 0,+, \geqslant)$. We then ask that this monoid be the positive part of a product of totally ordered Abelian groups, or more generally a monoid with gcd (this notion will be introduced in Chap. XI).

Symmetric Polynomials
We fix $n$ and $\mathbf{A}$ and we let $S_1, \ldots, S_n$ be the elementary symmetric polynomials at the $X_i$ ‘s in $\mathbf{A}\left[X_1, \ldots, X_n\right]$. They are defined by the equality
$$T^n+S_1 T^{n-1}+S_2 T^{n-2}+\cdots+S_n=\prod_{i=1}^n\left(T+X_i\right) .$$
We have $S_1=\sum_i X_i, S_n=\prod_i X_i, S_k=\sum_{J \in \mathcal{P}{k, n}} \prod{i \in J} X_i$. Recall the following well-known theorem (a proof is suggested in Exercise 3).

## 数学代写|交换代数代写Commutative Algebra代考|Weights, Homogeneous Polynomials

$$w\left(\underline{X}^{\underline{m}}\right)=\sum_i m_i w\left(X_i\right)$$

$$T^n+S_1 T^{n-1}+S_2 T^{n-2}+\cdots+S_n=\prod_{i=1}^n\left(T+X_i\right) .$$

## 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。

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

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## 数学代写|交换代数代写Commutative Algebra代考|Generalized Cramer Formula

We study in this subsection some generalizations of the usual Cramer formulas. We will exploit these in the following paragraphs.

For a matrix $A \in \mathbf{A}^{m \times n}$ we denote by $A_{\alpha, \beta}$ the matrix extracted on the rows $\alpha=\left{\alpha_1, \ldots, \alpha_r\right} \subseteq \llbracket 1 . . m \rrbracket$ and the columns $\beta=\left{\beta_1, \ldots, \beta_s\right} \subseteq \llbracket 1 . . n \rrbracket$.

Suppose that the matrix $A$ is of rank $\leqslant k$. Let $V \in \mathbf{A}^{m \times 1}$ be a column vector such that the bordered matrix $[A \mid V]$ is also of rank $\leqslant k$. Let us call $A_j$ the $j$-th column of $A$. Let $\mu_{\alpha, \beta}=\operatorname{det}\left(A_{\alpha, \beta}\right)$ be the minor of order $k$ of the matrix $A$ extracted on the rows $\alpha=\left{\alpha_1, \ldots, \alpha_k\right}$ and the columns $\beta=\left{\beta_1, \ldots, \beta_k\right}$. For $j \in \llbracket 1 . . k \rrbracket$ let $\nu_{\alpha, \beta, j}$ be the determinant of the same extracted matrix, except that the column $j$ has been replaced with the extracted column of $V$ on the rows $\alpha$. Then, we obtain for each pair $(\alpha, \beta)$ of multi-indices a Cramer identity:
$$\mu_{\alpha, \beta} V=\sum_{j=1}^k \nu_{\alpha, \beta, j} A_{\beta_j}$$
due to the fact that the rank of the bordered matrix $\left[A_{1 . . m, \beta} \mid V\right]$ is $\leqslant k$. This can be read as follows:
\begin{aligned} \mu_{\alpha, \beta} V & =\left[A_{\beta_1} \ldots A_{\beta_k}\right] \cdot\left[\begin{array}{c} \nu_{\alpha, \beta, 1} \ \vdots \ \nu_{\alpha, \beta, k} \end{array}\right] \ & =\left[A_{\beta_1} \ldots A_{\beta_k}\right] \cdot \operatorname{Adj}\left(A_{\alpha, \beta}\right) \cdot\left[\begin{array}{c} v_{\alpha_1} \ \vdots \ v_{\alpha_k} \end{array}\right] \ & =A \cdot\left(\mathrm{I}n\right){1 . . n, \beta} \cdot \operatorname{Adj}\left(A_{\alpha, \beta}\right) \cdot\left(\mathrm{I}m\right){\alpha, 1 . . m} \cdot V \end{aligned}
This leads us to introduce the following notation.

## 数学代写|交换代数代写Commutative Algebra代考|A Magic Formula

An immediate consequence of the Cramer’s identity (12) is the less usual identity (17) given in the following theorem. Similarly the equalities (18) and (19) easily result from (15) and (16).
5.14 Theorem Let $A \in \mathbf{A}^{m \times n}$ be a matrix of rank $k$. We thus have an equality $\sum_{\alpha \in \mathcal{P}{k, m}, \beta \in \mathcal{P}{k, n}} c_{\alpha, \beta} \mu_{\alpha, \beta}=1$. Let
$$B=\sum_{\alpha \in \mathcal{P}{k, m}, \beta \in \mathcal{P}{k, n}} c_{\alpha, \beta} \operatorname{Adj}_{\alpha, \beta}(A)$$

1. We have
$$A \cdot B \cdot A=A$$
Consequently $A B$ is a projection matrix of rank $k$ and the submodule $\operatorname{Im} A=$ $\operatorname{Im} A B$ is a direct summand in $\mathbf{A}^m$.
2. If $k=m$, then
$$A \cdot B=\mathrm{I}_m$$
3. If $k=n$, then
$$B \cdot A=\mathrm{I}n$$ The following identity, which we will not use in this work, is even more miraculous. 5.15 Proposition (Prasad and Robinson) With the assumptions and the notations of Theorem 5.14 , if we have $$\forall \alpha, \alpha^{\prime} \in \mathcal{P}{k, m}, \forall \beta, \beta^{\prime} \in \mathcal{P}{k, n} \quad c{\alpha, \beta} c_{\alpha^{\prime}, \beta^{\prime}}=c_{\alpha, \beta^{\prime}} c_{\alpha^{\prime}, \beta}$$
then
$$B \cdot A \cdot B=B$$

## 数学代写|交换代数代写Commutative Algebra代考|Generalized Cramer Formula

$$\mu_{\alpha, \beta} V=\sum_{j=1}^k \nu_{\alpha, \beta, j} A_{\beta_j}$$

\begin{aligned} \mu_{\alpha, \beta} V & =\left[A_{\beta_1} \ldots A_{\beta_k}\right] \cdot\left[\begin{array}{c} \nu_{\alpha, \beta, 1} \ \vdots \ \nu_{\alpha, \beta, k} \end{array}\right] \ & =\left[A_{\beta_1} \ldots A_{\beta_k}\right] \cdot \operatorname{Adj}\left(A_{\alpha, \beta}\right) \cdot\left[\begin{array}{c} v_{\alpha_1} \ \vdots \ v_{\alpha_k} \end{array}\right] \ & =A \cdot\left(\mathrm{I}n\right){1 . . n, \beta} \cdot \operatorname{Adj}\left(A_{\alpha, \beta}\right) \cdot\left(\mathrm{I}m\right){\alpha, 1 . . m} \cdot V \end{aligned}

## 数学代写|交换代数代写Commutative Algebra代考|A Magic Formula

5.14定理设$A \in \mathbf{A}^{m \times n}$为秩为$k$的矩阵。因此我们有一个等式$\sum_{\alpha \in \mathcal{P}{k, m}, \beta \in \mathcal{P}{k, n}} c_{\alpha, \beta} \mu_{\alpha, \beta}=1$。让
$$B=\sum_{\alpha \in \mathcal{P}{k, m}, \beta \in \mathcal{P}{k, n}} c_{\alpha, \beta} \operatorname{Adj}_{\alpha, \beta}(A)$$

$$A \cdot B \cdot A=A$$

$$J_{k, \ell}(\underline{X})=\left|\begin{array}{ll} \frac{\partial f}{\partial X_k}(\underline{X}) & \frac{\partial f}{\partial X_{\ell}}(\underline{X}) \ \frac{\partial g}{\partial X_k}(\underline{X}) & \frac{\partial g}{\partial X_{\ell}}(\underline{X}) \end{array}\right|$$

$$\sum_{1 \leqslant k<\ell \leqslant n} b_{k, \ell} j_{k, \ell}=1$$

## 数学代写|交换代数代写Commutative Algebra代考|The General Case

$$\mathbf{A}=\mathbf{R}\left[X_1, \ldots, X_n\right] /\left\langle f_1, \ldots, f_m\right\rangle=\mathbf{R}\left[x_1, \ldots, x_n\right]=\mathbf{R}[\underline{x}]$$

$$J(\underline{X})=\left[\begin{array}{ccc} \frac{\partial f_1}{\partial X_1}(\underline{X}) & \cdots & \frac{\partial f_1}{\partial X_n}(\underline{X}) \ \vdots & & \vdots \ \frac{\partial f_m}{\partial X_1}(\underline{X}) & \cdots & \frac{\partial f_m}{\partial X_n}(\underline{X}) \end{array}\right] .$$

$r \times r$的子条件表示$\mathbf{A}$的元素$b_{k_1, \ldots, k_r}^{i_1, \ldots, i_r}$的存在，使得
$$\sum_{1 \leqslant k_1<\cdots<k_r \leqslant n, 1 \leqslant i_1<\cdots<i_r \leqslant m} b_{k_1, \ldots, k_r}^{i_1, \ldots, i_r} j_{k_1, \ldots, k_r}^{i_1, \ldots, i_r}=1 .$$

$$\Omega_{\mathbf{A} / \mathbf{R}}=\left(\mathbf{A} \mathrm{d} x_1 \oplus \cdots \oplus \mathbf{A} \mathrm{d} x_n\right) /\left\langle\mathrm{d} f_1, \ldots, \mathrm{d} f_m\right\rangle \simeq \mathbf{A}^n / \operatorname{Im}^{\mathrm{t}} J$$

## 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。

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

## avatest™帮您通过考试

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## 数学代写|交换代数代写Commutative Algebra代考|Multivariate Factorization

In this chapter we show how to reduce the factorization of multivariate polynomials in $K\left[x_1, \ldots, x_n\right]$ to the case of one variable. The idea is similar to the reduction of the factorization in $\mathbb{Z}[x]$ to the factorization in $\mathbb{Z} / p[x]$. We choose a so-called main variable, say $x_n$ and a suitable point $a=\left(a_1, \ldots, a_{n-1}\right) \in K^{n-1}$. Let $\mathfrak{m}a \in K\left[x_1, \ldots, x{n-1}\right]$ be the maximal ideal corresponding to $a$. We factorize $f\left(a, x_n\right)$ in $K\left[x_n\right]=\left(K\left[x_1, \ldots, x_{n-1}\right] / \mathfrak{m}a\right)\left[x_n\right]$ and use Hensel lifting to lift the factors to $\left(K\left[x_1, \ldots, x{n-1}\right] / \mathfrak{m}_a^N\right)\left[x_n\right]$ for sufficiently large $N$. We choose (unique) representatives of these liftings and combine them to obtain the true factors. Let us start with an example.
Example B.6.1.
\begin{aligned} f= & x^4+(-z+3) x^3+\left(z^3+(y-3) z-y^2-13\right) x^2 \ & +\left(-z^4+\left(y^2+3 y+15\right) z+6\right) x \ & +y z^4+2 z^3+z\left(-y^3-15 y\right)-2 y^2-30 . \end{aligned}
We chose $x$ as main variable, $a=(0,0), \mathfrak{m}_a=\langle y, z\rangle$ and factorize $f(x, 0,0)$. We obtain
$$f(x, 0,0)=x^4+3 x^3-13 x^2+6 x-30=g_1 \cdot h_1$$
with $g_1=x^2+2$ and $h_1=x^2+3 x-15$.
We want to lift the factorization $f=g_1 h_1\left(\bmod \mathfrak{m}_a\right)$ to $f=g_i h_i\left(\bmod \mathfrak{m}_a^i\right)$ for increasing $i$ (Hensel lifting).
We have
\begin{aligned} f-g_1 h_1\left(\bmod \mathfrak{m}_a^2\right) & =-z x^3-3 z x^2+15 z x \ & =-z x \cdot h_1 . \end{aligned}
For the Hensel lifting we obtain
$$h_2=h_1 \text { and } g_2=g_1-z x=x^2-z x+2 .$$
In the next step we have
\begin{aligned} f-g_2 h_2 \bmod \mathfrak{m}_a^3 & =\left(y z-y^2\right) x^2+3 y z x-15 y z-2 y^2 \ & =y z g_2-y^2 \cdot h_2 \end{aligned}

## 数学代写|交换代数代写Commutative Algebra代考|Absolute Factorization

Let $K$ be a field of characteristic $0, \bar{K}$ its algebraic closure and assume we are able to compute the multivariate factorization over algebraic extensions of $K$ (our main example is $K=\mathbb{Q}$ ). In this chapter we explain how to compute the absolute factorization of a polynomial $f \in K\left[x_1, \ldots, x_n\right]$, that is, to compute the irreducible factors (and their multiplicities) of $f$ in $\bar{K}\left[x_1, \ldots, x_n\right]$. To solve this problem we may assume that $f$ is irreducible in $K\left[x_1, \ldots, x_n\right]$.

There exist several approaches to solve this problem (cf. [59], the part written by Chèze and Galligo, or [42]). We concentrate on the algorithm implemented by G. Lecerf in SingULAR.

The idea of this algorithm is to find an algebraic field extension $K(\alpha)$ of $K$ and a smooth point of the affine variety $V(f)$ in $K(\alpha)^n$. Then an (absolutely) irreducible factor of $f$ will be defined over $K(\alpha)$ which can be computed by using the factorization over $K(\alpha)$ described in Section B.5. The idea is based on the following theorem:

Theorem B.7.1. Let $f \in K\left[x_1, \ldots, x_n\right]$ be irreducible and $a \in K^n$ a smooth point of $V(f) \subseteq \bar{K}^n$. Then $f$ is absolutely irreducible, i.e. irreducible in $\bar{K}\left[x_1, \ldots, x_n\right]$

Proof. Let $f=f_1 \cdot \ldots \cdot f_t$ be the factorization of $f$ in $\bar{K}\left[x_1, \ldots, x_n\right]$. We may assume that $f_1(a)=0$. Assume that $t>1$. This implies that $f_i \notin$ $K\left[x_1, \ldots, x_n\right]$ for all $i$. Now $a$ being a smooth point of $V(f)$ implies that $f_i(a) \neq 0$ for $i>1$. We may choose $\alpha \in \bar{K}$ such that $f_i \in K(\alpha)\left[x_1, \ldots, x_n\right]$ for all $i$. Since $f_1 \notin K\left[x_1, \ldots, x_n\right]$ there exist $\sigma \in \operatorname{Gal}_K(K(\alpha))$ such that $\sigma\left(f_1\right) \neq f_1$.

But $\sigma(f)=f=\sigma\left(f_1\right) \cdot \ldots \cdot \sigma\left(f_t\right)$ implies that there is $i \neq 1$ such that $\sigma\left(f_1\right)=c \cdot f_i$ for some non-zero constant $c \in K(\alpha)$.
This implies $0=\sigma\left(f_1(a)\right)=c \cdot f_i(a)$ which is a contradiction. We obtain $t=1$ and $f$ is absolutely irreducible.
If we apply the theorem to $K(\alpha)$ we obtain:
Corollary B.7.2. Let $f \in K\left[x_1, \ldots, x_n\right]$ be irreducible, $\alpha \in \bar{K}$ and $a \in$ $K(\alpha)^n$ a smooth point of $V(f) \subseteq \bar{K}^n$, then at least one absolutely irreducible factor of $f$ is defined over $K(\alpha)$.

For $f$ irreducible it is not difficult to find $\alpha$ and a smooth point of $V(f)$ in $K(\alpha)^n$ (use Lemma B.6.8). We deduce that for irreducible $f \in K\left[x_1, \ldots, x_n\right]$, $\operatorname{deg}_{x_n}(f)>0, f\left(a, x_n\right)$ is squarefree for almost all $a \in K^{n-1}$.

## 数学代写|交换代数代写Commutative Algebra代考|Multivariate Factorization

\begin{aligned} f= & x^4+(-z+3) x^3+\left(z^3+(y-3) z-y^2-13\right) x^2 \ & +\left(-z^4+\left(y^2+3 y+15\right) z+6\right) x \ & +y z^4+2 z^3+z\left(-y^3-15 y\right)-2 y^2-30 . \end{aligned}

$$f(x, 0,0)=x^4+3 x^3-13 x^2+6 x-30=g_1 \cdot h_1$$

\begin{aligned} f-g_1 h_1\left(\bmod \mathfrak{m}_a^2\right) & =-z x^3-3 z x^2+15 z x \ & =-z x \cdot h_1 . \end{aligned}

$$h_2=h_1 \text { and } g_2=g_1-z x=x^2-z x+2 .$$

\begin{aligned} f-g_2 h_2 \bmod \mathfrak{m}_a^3 & =\left(y z-y^2\right) x^2+3 y z x-15 y z-2 y^2 \ & =y z g_2-y^2 \cdot h_2 \end{aligned}

## 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。