Posted on Categories:Homological Algebra, 同调代数, 数学代写

## 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|同调代数代写Homological Algebra代考|Degree One Group Cohomology

Already for $H^1(G, M)$ one needs some computation. An obvious starting point for the projective resolution of $R$ is
$$0 \longrightarrow I(R G) \stackrel{\iota}{\longrightarrow} R G \longrightarrow R \longrightarrow 0$$
and the rightmost mapping sends all $g \in G$ to 1 . The kernel of this mapping is $I(R G)$, the augmentation ideal, generated by $1-g \in R G$ for all $g \in G$, as a left ideal. Put
$$\operatorname{Der}(G, M)={f: G \longrightarrow M \mid f(g h)=g f(h)+f(g) \forall g, h \in G},$$
the derivations of $G$ with values in $M$, and
$$\operatorname{Inn} \operatorname{Der}(G, M):={f: G \longrightarrow M \mid \exists m \in M: f(g)=g m-m},$$
the inner derivations of $G$ with values in $M$. Then
$$H^1(G, M)=\operatorname{Der}(G, M) / \operatorname{Inn} \operatorname{Der}(G, M) .$$
Indeed, if
$$\cdots \longrightarrow P_2 \stackrel{\partial_1}{\longrightarrow} P_1 \stackrel{\partial_0}{\longrightarrow} R G \longrightarrow R$$
is a projective resolution, then
$$\operatorname{ker}\left(\operatorname{Hom}{R G}\left(\partial_1, M\right)\right)=\operatorname{Hom}{R G}(I(R G), M)$$
since the morphisms $\varphi: P_1 \longrightarrow M$ with $\varphi \circ \partial_1=0$ factor through
$$\operatorname{coker}\left(\partial_1\right)=\operatorname{ker}\left(\partial_0\right)=I(R G) .$$
Hence, since $g(h-1)=g h-g=(g h-1)-(g-1)$, we have
\begin{aligned} \operatorname{ker}\left(\operatorname{Hom}{R G}\left(\partial_1, M\right)\right)=& H m{R G}(I(R G), M) \ =&\left{\varphi \in \operatorname{Hom}_R(I(R G), M) \mid\right.\ &\forall g, h \in G: g \varphi(h-1)=\varphi(g h-1)-\varphi(g-1)} \ =&{f: G \longrightarrow M \mid\ &\forall g, h \in G: g \cdot f(h)=f(g h)-f(g)} \ =& \operatorname{Der}(G, M) . \end{aligned}

## 数学代写|同调代数代写Homological Algebra代考|Degree Two Group Cohomology

In order to compute $H^2(G, M)=E x t_{\mathbb{Z} G}^2(\mathbb{Z}, M)$ we need to give a projective resolution of the trivial module $\mathbb{Z}$ in a more explicit way, and in particular in a way that works for all groups $G$. For algebras in general this is the so-called bar resolution, which will be given in full detail and complete generality in Definition 3.6.4. However, up to degree 2 the resolution is sufficiently simple to be given immediately, in particular for group rings.
Using the fact that
$$0 \longrightarrow I(R G) \longrightarrow R G \longrightarrow R \longrightarrow 0$$
is an exact sequence where $R G$ is free, we need to find a projective $R G$-module $P$ and an epimorphism $P \longrightarrow I(R G)$. Let $P:=R G \otimes_R R G$ with the action of $R G$ by multiplication on the left term of the tensor product. This module is clearly projective, even free. We define
\begin{aligned} R G \otimes_R R G & \stackrel{\partial_0}{\longrightarrow} R G \ g_1 \otimes g_2 & \mapsto g_1\left(g_2-1\right) \end{aligned}
for all $g_1, g_2 \in G$. It is clear that this map is well-defined and surjective since $I(R G)$ is generated by $g-1$ for $g \in G$. Let $Q=R G \otimes_R R G \otimes_R R G$. This is a projective $R G$-module by multiplication on the left-most term. Let
\begin{aligned} R G \otimes_R R G \otimes_R R G & \stackrel{\partial_1}{\longrightarrow} R G \otimes_R R G \ g_1 \otimes g_2 \otimes g_3 \mapsto g_1 g_2 \otimes g_3-g_1 \otimes g_2 g_3+g_1 \otimes g_2 \end{aligned}

## 数学代写|同调代数代写同调代数代考|一次群同调

$$0 \longrightarrow I(R G) \stackrel{\iota}{\longrightarrow} R G \longrightarrow R \longrightarrow 0$$
，最右边的映射将所有$g \in G$发送到1。这个映射的内核是$I(R G)$，这是一个扩展理想，由$1-g \in R G$为所有$g \in G$生成，作为一个左理想。将
$$\operatorname{Der}(G, M)={f: G \longrightarrow M \mid f(g h)=g f(h)+f(g) \forall g, h \in G},$$

$$\operatorname{Inn} \operatorname{Der}(G, M):={f: G \longrightarrow M \mid \exists m \in M: f(g)=g m-m},$$

$$H^1(G, M)=\operatorname{Der}(G, M) / \operatorname{Inn} \operatorname{Der}(G, M) .$$

$$\cdots \longrightarrow P_2 \stackrel{\partial_1}{\longrightarrow} P_1 \stackrel{\partial_0}{\longrightarrow} R G \longrightarrow R$$

$$\operatorname{ker}\left(\operatorname{Hom}{R G}\left(\partial_1, M\right)\right)=\operatorname{Hom}{R G}(I(R G), M)$$

$$\operatorname{coker}\left(\partial_1\right)=\operatorname{ker}\left(\partial_0\right)=I(R G) .$$

\begin{aligned} \operatorname{ker}\left(\operatorname{Hom}{R G}\left(\partial_1, M\right)\right)=& H m{R G}(I(R G), M) \ =&\left{\varphi \in \operatorname{Hom}_R(I(R G), M) \mid\right.\ &\forall g, h \in G: g \varphi(h-1)=\varphi(g h-1)-\varphi(g-1)} \ =&{f: G \longrightarrow M \mid\ &\forall g, h \in G: g \cdot f(h)=f(g h)-f(g)} \ =& \operatorname{Der}(G, M) . \end{aligned}

## 数学代写|同调代数代写同调代数代考|度二群同调

$$0 \longrightarrow I(R G) \longrightarrow R G \longrightarrow R \longrightarrow 0$$

\begin{aligned} R G \otimes_R R G & \stackrel{\partial_0}{\longrightarrow} R G \ g_1 \otimes g_2 & \mapsto g_1\left(g_2-1\right) \end{aligned}
。很明显，这个映射是定义良好且满射的，因为$I(R G)$是由$g-1$为$g \in G$生成的。让$Q=R G \otimes_R R G \otimes_R R G$。这是一个投影$R G$模块，通过对最左边的项进行乘法运算。Let
\begin{aligned} R G \otimes_R R G \otimes_R R G & \stackrel{\partial_1}{\longrightarrow} R G \otimes_R R G \ g_1 \otimes g_2 \otimes g_3 \mapsto g_1 g_2 \otimes g_3-g_1 \otimes g_2 g_3+g_1 \otimes g_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。

Posted on Categories:Homological Algebra, 同调代数, 数学代写

## 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|同调代数代写Homological Algebra代考|Tensor Products

We are going to introduce a fundamental construction, the tensor product, which is absolutely necessary for higher algebra. In particular this is the abstract tool used for induced modules and Mackey’s formula, which in turn are the most important tools in the representation theory of finite groups over an arbitrary field.

Furthermore, tensor products are in some respect counterparts to homomorphisms, in a sense which can be made very precise. The precise formulation is given in Lemma 1.7.9 below, which will play an extremely important role in the subsequent material.

1.7.1 The Definition and Elementary Properties
The first concept to introduce is a free abelian group generated by a set.
Definition 1.7.1 An abelian group $A$ is free on a subset $S$ of $A$ if for every abelian group $B$ and every mapping $\varphi_S: S \longrightarrow B$ (as a set !) there is a unique homomorphism $\varphi: A \longrightarrow B$ such that the restriction $\left.\varphi\right|_S$ of $\varphi$ to $S$ equals $\varphi_S$.

Consider the abelian groups $\left(\mathbb{Z}^n,+\right)$ for any integer $n$. The group $\left(\mathbb{Z}^n,+\right)$ is a free abelian group on the set
$${(1,0,0, \ldots, 0),(0,1,0, \ldots, 0), \ldots,(0,0, \ldots, 0,1)}$$
in the definition below. Indeed, the above set is a basis of $\mathbb{Z}^n$ in the sense that for any abelian group $B$ and any $n$ elements $b_1, b_2, \cdots, b_n$ of $B$ there is a unique homomorphism of abelian groups $\varphi: \mathbb{Z}^n \longrightarrow B$ such that
$$\begin{gathered} \varphi(1,0,0, \ldots, 0)=b_1 \ \varphi(0,1,0, \ldots, 0)=b_2 \ \ldots \ldots \ldots \ldots \ldots \ldots . \ \varphi(0,0, \ldots, 0,1)=b_n \end{gathered}$$
$\varphi\left(a_1, a_2, \ldots, a_n\right)=\sum_{i=1}^n a_i b_i$ is a group homomorphism and is the unique one with the above properties.

Suppose $A$ is a free abelian group on a set $S_A$, and suppose $B$ is another free abelian group on a set $S_B$. If $S_A$ and $S_B$ are of the same cardinality (i.e. there is a

bijection $\beta: S_A \longrightarrow S_B$ ) then $A$ and $B$ are isomorphic. Indeed, $\beta$ defines a unique homomorphism of groups $\widehat{\beta}: A \longrightarrow B$ restricting to $\beta$ on $S_A$, and $\beta^{-1}$ defines a unique homomorphism of groups $\widehat{\beta^{-1}}: A \longrightarrow B$ restricting to $\beta$ on $S_B$. Now, the identity on $S_A$ is the unique group homomorphism $A \longrightarrow A$ restricting to the identity on $S_A$. But, $\widehat{\beta^{-1}} \circ \widehat{\beta}$ is a group homomorphism restricting to $\beta^{-1} \circ \beta=i d_{S_A}$ on $S_A$. Therefore, by the unicity, $\widehat{\beta^{-1}} \circ \widehat{\beta}=i d_A$. Analogously, $\widehat{\beta} \circ \widehat{\beta^{-1}}=i d_A$. Hence, $\widehat{\beta}$ is an isomorphism.

## 数学代写|同调代数代写Homological Algebra代考|Immediate Applications for Group Rings

If $E$ is an extension field of $K, A$ is a $K$-algebra and $M$ is an $A$-module, then by Lemma 1.7.12 $E \otimes_K A$ is again an algebra. Moreover, by Lemma 1.7.8 the tensor product $E \otimes_K A$ is an $E$-vector space. The multiplication inside $E \otimes_K A$ implies that the elements $e \otimes 1$ for $e \in E$ are all central, and hence $E \otimes_K A$ is an $E$-algebra. The module $E \otimes_K M$ is an $E \otimes_K A$-module by Lemma 1.7.14. This procedure is called a “change of rings”, meaning change of base rings. We observe that the discussion in Sect. 1.4.3 is precisely a down-to-earth description of this abstract “change of rings”.
The second occasion where tensor products are used is a generalisations of this: induction of modules. We start with the observation that given a commutative ring $R$, an $R$-algebra $A$ and an $R$-subalgebra $B$, the ring $B$ acts on $A$ by right multiplication. In other words, restricting the action of $A$ on the regular right $A$-module $A$ to $B$ gives $A$ the structure of a right $B$-module. Also, $A$ can be regarded as a regular left $A$-module. Both actions commute. That is, $A$ is an $A-B$-bimodule if we define $a \cdot x \cdot b:=a x b$ for all $a, x \in A$ and $b \in B$.

Definition 1.7.20 Let $G$ be a group, let $R$ be a commutative ring and let $H$ be a subgroup of $G$. Then for every $R H$-module $M$ one defines the induced module $M \uparrow \uparrow_H^G$ to be $R G \otimes_{R H} M$ where for all $g \in G$ and all $x \in R G, m \in M$ we define $g \cdot(x \otimes m):=(g x) \otimes m .$
The induced module is of extreme importance in representation theory.
Remark 1.7.21 Tensor products also appear in more sophisticated structures, socalled Hopf-algebras. For a very brief introduction see Sect.6.2.1 or e.g. Montgomery’s book [10]. Group rings are Hopf algebras. For a commutative ring $K$ and a group $G$ as well as two $K G$-modules $M_1$ and $M_2$ we get that $M_1 \otimes_K M_2$ is a also $K G$-module by putting $g \cdot\left(m_1 \otimes m_2\right):=\left(g m_1 \otimes g m_2\right)$ for all $g \in G$ and $m_1 \in M_1$, $m_2 \in M_2$. Indeed, for all $g \in G$ the mapping
\begin{aligned} g \cdot: M_1 \times M_2 & \longrightarrow M_1 \otimes_K M_2 \ \left(m_1, m_2\right) & \mapsto g m_1 \otimes g m_2 \end{aligned}
is $K$-balanced and so it induces a mapping
$$\text { g. }: M_1 \otimes_K M_2 \longrightarrow M_1 \otimes_K M_2$$

$m_1 \otimes m_2 \mapsto g m_1 \otimes g m_2$
which in turn induces a $K$-linear action of $G$ on $M_1 \otimes_K M_2$, since $1_G \cdot=i d_{M_1 \otimes_K M_2}$ and $\left(g_1 g_2\right) \cdot=\left(g_{1^*}\right) \circ\left(g_2 \cdot\right)$ for all $g_1, g_2 \in G$ as is immediately checked.

However, a warning should be given here. Elements $g_1+g_2$ in $K G$ do not act diagonally on $M_1 \otimes_K M_2$. In fact, for $m_1 \in M_1$ and $m_2 \in M_2$ we have
\begin{aligned} \left(g_1+g_2\right) \cdot\left(m_1 \otimes m_2\right) &=g_1 \cdot\left(m_1 \otimes m_2\right)+g_2 \cdot\left(m_1 \otimes m_2\right) \ &=\left(g_1 m_1 \otimes g_1 m_2\right)+\left(g_2 m_1 \otimes g_2 m_2\right) \end{aligned}
whereas
\begin{aligned} \left(g_1+g_2\right) m_1 \otimes\left(g_1+g_2\right) m_2=&\left(g_1 m_1 \otimes g_1 m_2\right)+\left(g_1 m_1 \otimes g_2 m_2\right) \ &+\left(g_2 m_1 \otimes g_1 m_2\right)+\left(g_2 m_1 \otimes g_2 m_2\right) . \end{aligned}
We now continue to examine induced modules.

## 数学代写|同调代数代写同质代数代考|张量积

$${(1,0,0, \ldots, 0),(0,1,0, \ldots, 0), \ldots,(0,0, \ldots, 0,1)}$$

$$\begin{gathered} \varphi(1,0,0, \ldots, 0)=b_1 \ \varphi(0,1,0, \ldots, 0)=b_2 \ \ldots \ldots \ldots \ldots \ldots \ldots . \ \varphi(0,0, \ldots, 0,1)=b_n \end{gathered}$$
$\varphi\left(a_1, a_2, \ldots, a_n\right)=\sum_{i=1}^n a_i b_i$是一个群同态，并且是具有上述性质的唯一的一个

## 数学代写|同调代数代写Homological Algebra代考|群环的直接应用

\begin{aligned} g \cdot: M_1 \times M_2 & \longrightarrow M_1 \otimes_K M_2 \ \left(m_1, m_2\right) & \mapsto g m_1 \otimes g m_2 \end{aligned}

$$\text { g. }: M_1 \otimes_K M_2 \longrightarrow M_1 \otimes_K M_2$$

$m_1 \otimes m_2 \mapsto g m_1 \otimes g m_2$
，这反过来诱导了$G$对$M_1 \otimes_K M_2$的$K$ -线性动作，因为$1_G \cdot=i d_{M_1 \otimes_K M_2}$和$\left(g_1 g_2\right) \cdot=\left(g_{1^*}\right) \circ\left(g_2 \cdot\right)$对于所有$g_1, g_2 \in G$立即被检查

\begin{aligned} \left(g_1+g_2\right) \cdot\left(m_1 \otimes m_2\right) &=g_1 \cdot\left(m_1 \otimes m_2\right)+g_2 \cdot\left(m_1 \otimes m_2\right) \ &=\left(g_1 m_1 \otimes g_1 m_2\right)+\left(g_2 m_1 \otimes g_2 m_2\right) \end{aligned}

\begin{aligned} \left(g_1+g_2\right) m_1 \otimes\left(g_1+g_2\right) m_2=&\left(g_1 m_1 \otimes g_1 m_2\right)+\left(g_1 m_1 \otimes g_2 m_2\right) \ &+\left(g_2 m_1 \otimes g_1 m_2\right)+\left(g_2 m_1 \otimes g_2 m_2\right) . \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:Homological Algebra, 同调代数, 数学代写

## 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|同调代数代写Homological Algebra代考|Noetherian and Artinian Objects

Being a finite dimensional algebra over a field often is too strong a condition. The appropriate concept is that of a Noetherian or artinian module, which we will introduce now. These concepts are by far less restrictive, but are sufficiently strong to allow the most important results for finite dimensional algebras, at least those properties that are interesting to us.

Definition 1.3.1 Let $K$ be a commutative ring, let $A$ be a $K$-algebra and let $M$ be an $A$-module.

$M$ is said to be Noetherian if, whenever there is a sequence
$$M_1 \subseteq M_2 \subseteq M_3 \subseteq \cdots \subseteq M$$
of $A$-submodules of $M$, then there is an $n_0 \in \mathbb{N}$ such that $M_n=M_{n_0}$ for all $n \geq n_0$.

$M$ is said to be artinian if, whenever there is a sequence
$$M \supseteq M_1 \supseteq M_2 \supseteq M_3 \supseteq \ldots$$
of $A$-submodules of $M$, then there is an $n_0 \in \mathbb{N}$ such that $M_n=M_{n_0}$ for all $n \geq n_0$.

A is said to be left (right) Noetherian if the left (right) regular module is Noetherian.

$A$ is said to be left (right) artinian if the left (right) regular module is artinian.
Of course, if $A$ is an algebra over a field $K$, then any $A$-module of finite dimension as a $K$-vector space is Noetherian and artinian.

Example 1.3.2 The ring of integers $\mathbb{Z}$ is Noetherian since for any ideal $I=n \mathbb{Z}$ of $\mathbb{Z}$, an ideal $J=m \mathbb{Z}$ contains $I$ if and only if $m$ divides $n$. There are only finitely many divisors of $n$, and the statement is proven.
The ring of integers is not artinian, since the sequence of ideals
$$\mathbb{Z} \supseteq 2 \mathbb{Z} \supseteq 4 \mathbb{Z} \supseteq 8 \mathbb{Z} \supseteq \ldots$$
obviously is not finite.
The properties in the following lemmas are essential and are the reason for the importance of the notions Noetherian and artinian.

## 数学代写|同调代数代写Homological Algebra代考|Wedderburn and Krull-Schmidt

The main reason why Maschke’s result Theorem 1.2.8 is one of the most fundamental in the representation theory of finite groups is that the structure theory of finite dimensional semisimple algebras is known in great detail. We shall now develop this structure.
The Krull-Schmidt Theorem
For the moment we do not know anything about the unicity of a decomposition of a semisimple module into its factors. This is the very important Krull-Schmidt theorem.

It holds more generally, replacing simple modules by indecomposable modules as factors and semisimple algebras by general finite dimensional algebras.

Definition 1.4.1 A non-zero $A$-module $M$ is called indecomposable if whenever $M \simeq N \oplus L$, then either $N=0$ or $L=0$. Modules which are not indecomposable are decomposable.

We have seen that (by definition) semisimple indecomposable modules are simple. In general, however, indecomposable modules need not be simple.

Example 1.4.2 Let $K$ be a field and let $A=K[X] /\left(X^2\right)$ be the so-called ring of dual numbers. $A$ is a $K$-algebra of dimension 2 over $K$, and the regular module is indecomposable but not semisimple. Indeed, the only non-zero proper ideal of $A$ is $X \cdot K[X] /\left(X^2\right)$

## 数学代写|同调代数代写同构代数代考|Noetherian和Artinian Objects

$M$被认为是Noetherian if，当$M$的$A$ -子模块中有一个序列
$$M_1 \subseteq M_2 \subseteq M_3 \subseteq \cdots \subseteq M$$
，那么有一个$n_0 \in \mathbb{N}$，这样$M_n=M_{n_0}$对于所有$n \geq n_0$ .

$M$被认为是artinian if，无论在$M$的$A$ -子模块中有一个序列
$$M \supseteq M_1 \supseteq M_2 \supseteq M_3 \supseteq \ldots$$
，那么就有一个$n_0 \in \mathbb{N}$，这样$M_n=M_{n_0}$对于所有$n \geq n_0$ 如果左(右)正则模块是Noetherian，则A被称为左(右)Noetherian

$A$如果左(右)正则模块是artiinian的，则被称为左(右)artiinian的。当然，如果$A$是$K$字段上的代数，那么任何有限维数为$K$ -向量空间的$A$ -模块都是Noetherian和artinian 例1.3.2整数环$\mathbb{Z}$是Noetherian的，因为对于$\mathbb{Z}$的任意理想$I=n \mathbb{Z}$，当且仅当$m$除$n$时，理想$J=m \mathbb{Z}$包含$I$。$n$的除数是有限的，并且证明了这个说法。由于理想序列
$$\mathbb{Z} \supseteq 2 \mathbb{Z} \supseteq 4 \mathbb{Z} \supseteq 8 \mathbb{Z} \supseteq \ldots$$

## 数学代写|同调代数代写同源代数代考|Wedderburn and Krull-Schmidt

Maschke结果定理1.2.8是有限群表示理论中最基本的定理之一，其主要原因是有限维半单代数的结构理论被非常详细地了解。我们现在将发展这个结构。Krull-Schmidt定理目前我们还不知道半单模分解为其因子的唯一性。这是非常重要的Krull-Schmidt定理， 更一般地，用不可分解模代替简单模作为因子，用一般有限维代数代替半单代数 一个非零的$A$ -模块$M$被称为不可分解的，如果无论何时$M \simeq N \oplus L$，那么$N=0$或$L=0$。不是不可分解的模块是可分解的 我们已经看到(根据定义)半单不可分解模块是简单的。然而，一般情况下，不可分解模块不需要是简单的 例1.4.2设$K$为字段，设$A=K[X] /\left(X^2\right)$为所谓的双数字环。$A$是$K$上2维的$K$ -代数，并且常规模块是不可分解的，但不是半简单的。实际上，$A$的唯一非零固有理想是$X \cdot K[X] /\left(X^2\right)$

## 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。