### 数学代考|代数几何代写algebraic geometry代考|The Category of Topological Spaces

statistics-lab™ 为您的留学生涯保驾护航 在代写代数几何algebraic geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数几何algebraic geometry代写方面经验极为丰富，各种代写代数几何algebraic geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|代数几何代写algebraic geometry代考|The Category of Algebraic Varieties

Recall that at this moment, our definition of an algebraic variety includes affine, quasiaffine, projective and quasiprojective varieties, defined in Sects. 1.3.1 and 1.3.2. These are the objects of the category of algebraic varieties. Following our general guiding principle, morphisms of algebraic varieties $f: X \rightarrow Y$ are the mappings which preserve the structure.

The structure consists of the Zariski topology, and regular functions. It is correct to say that morphisms of varieties are those mappings which preserve topology and regular functions.
In more detail, then, morphisms of algebraic varieties
$$f: X \rightarrow Y$$
are maps which

1. are continuous with respect to the Zariski topology
2. have the property that if $g: U \rightarrow \mathbb{C}$ is a regular function, then the composition
$$g \circ f: f^{-1}(U) \rightarrow \mathbb{C}$$

or, more precisely,
$$\left.g \circ f\right|_{f^{-1}(U)}: f^{-1}(U) \rightarrow \mathbb{C},$$
is a regular function. Note the inverse image in both formulas. The extra notation in the second formula means the restriction of a function.

Note that in particular, by the second property, a morphism of varieties $f: X \rightarrow Y$ specifies (we sometimes say: induces) a homomorphism of rings
$$\mathbb{C}[f]: \mathbb{C}[Y] \rightarrow \mathbb{C}[X]$$
In fact, both rings contain $\mathbb{C}$ (thought of as constant functions), and the homomorphism $\mathbb{C}[f]$ fixes $\mathbb{C}$, i.e. satisfies
$$\mathbb{C}f=\lambda \text { for } \lambda \in \mathbb{C}$$
Commutative rings $R$ with a homomorphism of rings
$$A \rightarrow R$$
for some other commutative ring $A$ are called (commutative) A-algebras. Therefore, $C[f]$ is a homomorphism of $\mathbb{C}$-algebras. In general, a homomorphism of $A$-algebras $R \rightarrow R^{\prime}$ is defined as a homomorphism of rings which commutes with the homomorphisms from $A$ to $R, R^{\prime}$.

## 数学代考|代数几何代写algebraic geometry代考|The Morphisms into an Affine Variety

An easy but powerful theorem states that a morphism in the category of varieties
$$f: X \rightarrow Y$$
where $Y$ is affine is characterized by the induced homomorphism of $\mathbb{C}$-algebras
$$\mathbb{C}[X] \leftarrow \mathbb{C}[Y]$$
(note the reversal of the arrow, called contravariance).

This means, in more detail, that for every homomorphism of rings (2.3.2), there exists a unique morphism of varieties $(2.3 .1)$ which induces it, provided that $Y$ is affine. In particular:

The category of affine varieties over $\mathbb{C}$ and morphisms of varieties is equivalent to the opposite category of the category of finitely generated $\mathbb{C}$-algebras and homomorphisms of C-algebras.

To see why $(2.3 .1)$ and $(2.3 .2)$ are equivalent for $Y$ affine, note that the passage from (2.3.1) to $(2.3 .2)$ is immediate from the definition. On the other hand, given a homomorphism
$$\mathbb{C}\left[x_{1}, \ldots, x_{n}\right] / I(Y) \rightarrow \mathbb{C}[X]$$
we can take the images of the generators $x_{1}, \ldots, x_{n}$ as coordinates of a mapping from $X$ to $\mathrm{A}_{\mathrm{C}}^{n}$, which lands in $Y$. It is readily checked that this is a morphism of varieties, and that both passages between (2.3.1) and (2.3.2) are inverse to each other (although note that in our current setting, there are several cases for $X$ to consider!).

Roughly speaking, we think of affine varieties as those which have “enough regular functions.” From this point of view, they are the opposite of projective varieties: The only algebraic variety which is both affine and projective is a single point (and in some definitions, the empty set, but our definition of irreducibility excludes the empty set, so we do not count it).

## 数学代考|代数几何代写algebraic geometry代考|Quasiaffine Varieties which are not Isomorphic to Affine Varieties

The theorem described at the beginning of Sect. $2.3$ can be useful in deciding which varieties are isomorphic to affine varieties. For example, note that in $\mathrm{A}{\mathbb{C}}^{n}$, $${(0, \ldots, 0)}=Z\left(x{1}, \ldots, x_{n}\right)$$
( $Z$ denotes the set of zeros, see Sect. 1.1.2). By (1.4.6), (1.4.2), for $n \geq 2$, we have
\begin{aligned} &\mathbb{C}\left[\mathbb{A}{\mathbb{C}}^{n} \backslash{(0, \ldots, 0)}\right]= \ &=\mathbb{C}\left[x{1}, \ldots, x_{n}\right]\left[x_{1}^{-1}\right] \cap \cdots \cap \mathbb{C}\left[x_{1}, \ldots, x_{n}\right]\left[x_{n}^{-1}\right]= \ &=\mathbb{C}\left[x_{1}, \ldots, x_{n}\right]=\mathbb{C}\left[\mathbb{A}{\mathbb{C}}^{n}\right] \end{aligned} Since we know that $\mathrm{A}{\mathbb{C}}^{n}$ is affine, this means that
$$\mathrm{A}{\mathbb{C}}^{n} \backslash{(0, \ldots 0)}$$ is not affine for $n \geq 2$ : its inclusion into $\mathrm{A}{\mathrm{C}}^{n}$ induces an isomorphism of rings of regular function, but is not an isomorphism of varieties (it is not onto on points), so if both varieties were isomorphic to affine varieties, it would contradict the theorem at the beginning of Sect. 2.3).

## 数学代考|代数几何代写algebraic geometry代考|The Category of Algebraic Varieties

F:X→是

1. 关于 Zariski 拓扑是连续的
2. 拥有如果G:在→C是正则函数，则组成
G∘F:F−1(在)→C

G∘F|F−1(在):F−1(在)→C,

C[F]:C[是]→C[X]

$$\mathbb{C} f =\lambda \text { for } \lambda \in \mathbb{C} C这米米在吨一种吨一世在和r一世nGsR在一世吨H一种H这米这米这rpH一世s米这Fr一世nGs 其他交换环的 \rightarrow R$$

## 数学代考|代数几何代写algebraic geometry代考|The Morphisms into an Affine Variety

F:X→是

C[X]←C[是]
（注意箭头的反转，称为逆变）。

C[X1,…,Xn]/一世(是)→C[X]

## 数学代考|代数几何代写algebraic geometry代考|Quasiaffine Varieties which are not Isomorphic to Affine Varieties

Sect 开头描述的定理。2.3可用于确定哪些变体与仿射变体同构。例如，请注意在一种Cn,(0,…,0)=从(X1,…,Xn)
( 从表示零的集合，参见 Sect. 1.1.2)。由（1.4.6），（1.4.2），对于n≥2， 我们有
C[一种Cn∖(0,…,0)]= =C[X1,…,Xn][X1−1]∩⋯∩C[X1,…,Xn][Xn−1]= =C[X1,…,Xn]=C[一种Cn]既然我们知道一种Cn是仿射的，这意味着

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。