如果你也在 怎样代写交换代数Commutative Algebra 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。交换代数Commutative Algebra是计划的局部研究中的主要技术工具。对不一定是换元的环的研究被称为非换元代数；它包括环理论、表示理论和巴拿赫代数的理论。
交换代数Commutative Algebra换元代数本质上是对代数数论和代数几何中出现的环的研究。在代数理论中，代数整数的环是Dedekind环，因此它构成了一类重要的换元环。与模块化算术有关的考虑导致了估值环的概念。代数场扩展对子环的限制导致了积分扩展和积分封闭域的概念，以及估值环扩展的公理化概念。
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数学代写|交换代数代写commutative algebra代考|Newton’s Method in Algebra
Let $\mathbf{k}$ be a ring and $f_1, \ldots, f_s \in \mathbf{k}[X]=\mathbf{k}\left[X_1, \ldots, X_n\right]$. The Jacobian matrix of the system is the matrix
$$
\mathrm{JAC}{X_1, \ldots, X_n}\left(f_1, \ldots, f_s\right)=\left(\frac{\partial f_i}{\partial X_j}\right){i \in \llbracket 1 . . s \rrbracket, j \in \llbracket 1 . . n \rrbracket} \in \mathbf{k}[X]^{s \times n} .
$$
It is also denoted by $\mathrm{JAC}_{X}(f)$ or $\mathrm{JAC}(f)$. It is visualized as follows
If $s=n$, we denote by $\operatorname{Jac}{X}(f)$ or $\operatorname{Jac}{X_1, \ldots, X_n}\left(f_1, \ldots, f_n\right)$ or $\operatorname{Jac}(f)$ the Jacobian of the system $(f)$, i.e. the determinant of the Jacobian matrix.
In analysis Newton’s method to approximate a root of a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ is the following. Starting from a point $x_0$ “near a root,” at which the derivative is “far from 0 “, we construct a series $\left(x_m\right){m \in \mathbb{N}}$ by induction by letting $$ x{m+1}=x_m-\frac{f\left(x_m\right)}{f^{\prime}\left(x_m\right)} .
$$
The method can be generalized for a system of $p$ equations with $p$ unknowns. A solution of such a system is a zero of a function $f: \mathbb{R}^p \rightarrow \mathbb{R}^p$. We apply “the same formula” as above
$$
x_{m+1}=x_m-f^{\prime}\left(x_m\right)^{-1} \cdot f\left(x_m\right),
$$
where $f^{\prime}(x)$ is the differential (the Jacobian matrix) of $f$ at the point $x \in \mathbb{R}^p$, which must be invertible in a neighborhood of $x_0$.
数学代写|交换代数代写commutative algebra代考|Definition, Changing Generator Set
A finitely presented module is an A-module $M$ given by a finite number of generators and relations. Therefore it is a module with a finite generator set having a finitely generated syzygy module. Equivalently, it is a module $M$ isomorphic to the cokernel of a linear map
$$
\gamma: \mathbf{A}^m \longrightarrow \mathbf{A}^q
$$
The matrix $G \in \mathbf{A}^{q \times m}$ of $\gamma$ has as its columns a generator set of the syzygy module between the generators $g_i$ which are the images of the canonical base of $\mathbf{A}^q$ by the surjection $\pi: \mathbf{A}^q \rightarrow M$. Such a matrix is called a presentation matrix of the module $M$ for the generator set $\left(g_1, \ldots, g_q\right)$. This translates into
- $\left[g_1 \cdots g_q\right] G=0$, and
- every syzygy between the $g_i$ ‘s is a linear combination of the columns of $G$, i.e.: if $\left[g_1 \cdots g_q\right] C=0$ with $C \in \mathbf{A}^{q \times 1}$, there exists a $C^{\prime} \in \mathbf{A}^{m \times 1}$ such that $C=G C^{\prime}$.
Examples
1) A free module of rank $k$ is a finitely presented module presented by a matrix column formed of $k$ zeros. ${ }^1$ More generally every simple matrix is the presentation matrix of a free module of finite rank.
2) Recall that a finitely generated projective module is a module $P$ isomorphic to the image of a projection matrix $F \in \mathbb{M}_n(\mathbf{A})$ for a specific integer $n$. Since $\mathbf{A}^n=\operatorname{Im}(F) \oplus \operatorname{Im}\left(\mathrm{I}_n-F\right)$, we obtain $P \simeq \operatorname{Coker}\left(\mathrm{I}_n-F\right)$. This shows that every finitely generated projective module is finitely presented.
3) Let $\varphi: V \rightarrow V$ be an endomorphism of a finite-dimensional vector space over a discrete field $\mathbf{K}$. Consider $V$ as a $\mathbf{K}[X]$-module with the following external law
$$
\left{\begin{aligned}
\mathbf{K}[X] \times V & \rightarrow V \
(P, u) & \mapsto P \cdot u:=P(\varphi)(u) .
\end{aligned}\right.
$$
Let $\left(u_1, \ldots, u_n\right)$ be a basis of $V$ as a $\mathbf{K}$-vector space and $A$ be the matrix of $\varphi$ with respect to this basis. Then we can show that a presentation matrix of $V$ as a $\mathbf{K}[X]$-module for the generator set $\left(u_1, \ldots, u_n\right)$ is the matrix $X \mathrm{I}_n-A$ (see Exercise 3).
交换代数代考
数学代写|交换代数代写commutative algebra代考|Newton’s Method in Algebra
让$\mathbf{k}$成为一个环，$f_1, \ldots, f_s \in \mathbf{k}[X]=\mathbf{k}\left[X_1, \ldots, X_n\right]$。系统的雅可比矩阵就是这个矩阵
$$
\mathrm{JAC}{X_1, \ldots, X_n}\left(f_1, \ldots, f_s\right)=\left(\frac{\partial f_i}{\partial X_j}\right){i \in \llbracket 1 . . s \rrbracket, j \in \llbracket 1 . . n \rrbracket} \in \mathbf{k}[X]^{s \times n} .
$$
它也用$\mathrm{JAC}_{X}(f)$或$\mathrm{JAC}(f)$表示。它可视化如下
如果$s=n$，我们用$\operatorname{Jac}{X}(f)$或$\operatorname{Jac}{X_1, \ldots, X_n}\left(f_1, \ldots, f_n\right)$或$\operatorname{Jac}(f)$表示系统$(f)$的雅可比矩阵，即雅可比矩阵的行列式。
在分析中，牛顿近似可微函数$f: \mathbb{R} \rightarrow \mathbb{R}$的根的方法如下。从一个“接近根”的点$x_0$开始，在这个点导数“远离0”，我们用归纳法构造一个级数$\left(x_m\right){m \in \mathbb{N}}$，让$$ x{m+1}=x_m-\frac{f\left(x_m\right)}{f^{\prime}\left(x_m\right)} .
$$
该方法可推广到含有$p$未知数的$p$方程组。这样一个系统的解是一个函数$f: \mathbb{R}^p \rightarrow \mathbb{R}^p$的零。我们采用与上述“相同的公式”
$$
x_{m+1}=x_m-f^{\prime}\left(x_m\right)^{-1} \cdot f\left(x_m\right),
$$
其中$f^{\prime}(x)$是$f$在$x \in \mathbb{R}^p$点的微分(雅可比矩阵)，在$x_0$的邻域内必须是可逆的。
数学代写|交换代数代写commutative algebra代考|Definition, Changing Generator Set
有限表示的模块是由有限数量的生成器和关系给出的A模块$M$。因此，它是一个具有有限生成的syzygy模块的有限发电机组模块。等价地，它是一个与线性映射的核同构的模块$M$
$$
\gamma: \mathbf{A}^m \longrightarrow \mathbf{A}^q
$$
$\gamma$的矩阵$G \in \mathbf{A}^{q \times m}$的列是生成器$g_i$之间的syzygy模块的生成集，这些生成器是通过射$\pi: \mathbf{A}^q \rightarrow M$得到的$\mathbf{A}^q$的规范基的图像。这样的矩阵称为发电机组$\left(g_1, \ldots, g_q\right)$模块$M$的表示矩阵。这转化为
$\left[g_1 \cdots g_q\right] G=0$，和
$g_i$之间的每一个协同都是$G$的列的线性组合，即:如果$\left[g_1 \cdots g_q\right] C=0$与$C \in \mathbf{A}^{q \times 1}$，则存在一个$C^{\prime} \in \mathbf{A}^{m \times 1}$，使得$C=G C^{\prime}$。
示例
1)秩为$k$的自由模块是由$k$个零组成的矩阵列表示的有限呈现模块。${ }^1$更一般地说，每个简单矩阵都是有限秩自由模的表示矩阵。
2)回想一下，一个有限生成的投影模块是一个模块$P$同构于一个特定整数$n$的投影矩阵$F \in \mathbb{M}_n(\mathbf{A})$的像。由于$\mathbf{A}^n=\operatorname{Im}(F) \oplus \operatorname{Im}\left(\mathrm{I}_n-F\right)$，我们得到$P \simeq \operatorname{Coker}\left(\mathrm{I}_n-F\right)$。这说明每一个有限生成的投影模都是有限表示的。
3)设$\varphi: V \rightarrow V$为离散场$\mathbf{K}$上有限维向量空间的自同态。将$V$视为具有以下外部律的$\mathbf{K}[X]$ -模块
$$
\left{\begin{aligned}
\mathbf{K}[X] \times V & \rightarrow V \
(P, u) & \mapsto P \cdot u:=P(\varphi)(u) .
\end{aligned}\right.
$$
设$\left(u_1, \ldots, u_n\right)$是$V$作为$\mathbf{K}$向量空间的一组基$A$是$\varphi$关于这个基的矩阵。然后我们可以证明，作为发电机组$\left(u_1, \ldots, u_n\right)$的$\mathbf{K}[X]$ -模块的表示矩阵$V$是矩阵$X \mathrm{I}_n-A$(参见练习3)。
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