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数学代写|交换代数代写Commutative Algebra代考|Spectrum and Affine Schemes

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数学代写|交换代数代写Commutative Algebra代考|Spectrum and Affine Schemes

Abstract algebraic geometry, as introduced by Grothendieck, is a far reaching generalization of classical algebraic geometry. One of the main points is that it allows the application of geometric methods to arbitrary commutative rings, for example, to the ring $\mathbb{Z}$. Thus, geometric methods can be applied to number theory, creating a new discipline called arithmetic geometry.

However, even for problems in classical algebraic geometry, the abstract approach has turned out to be very important.

For example, for polynomial rings over an algebraically closed field, affine schemes provide more structure than classical algebraic sets. In a systematic manner, the abstract approach allows nilpotent elements in the coordinate ring. This has the advantage of understanding and describing much better “dynamic aspects” of a variety, since nilpotent elements occur naturally in the fibre of a morphism, that is, when a variety varies in an algebraic family.
The abstract approach to algebraic geometry has, however, the disadvantage that it is often far away from intuition, although a geometric language is used. A scheme has many more points than a classical variety, even a lot of non-closed points. This fact, although against any “classical” geometric feeling, has, on the other hand, the effect that the underlying topological space of a scheme carries more information. For example, the abstract Nullstellensatz, which is formally the same as Hilbert’s Nullstellensatz, holds without any assumption. However, since the geometric assumptions are much stronger than in the classical situation (we make assumptions on all prime ideals containing an ideal, not only on the maximal ideals), the abstract Nullstellensatz is more a remark than a theorem and Hilbert’s Nullstellensatz is not a consequence of the abstract one. Nevertheless, the formal coincidence makes the formulation of geometric results in the language of schemes much smoother, and the relation between algebra and geometry is, even for arbitrary rings, as close as it is for classical algebraic sets defined by polynomials over an algebraically closed field.

At the end of Section A.5, we shall show how results about algebraic sets can, indeed, be deduced from results about schemes (in a functorial manner).

Definition A.3.1. Let $A$ be a ring. Then
$$\operatorname{Spec}(A):={P \subset A \mid P \text { is a prime ideal }}$$
is called the (prime) spectrum of $A$, and
$$\operatorname{Max}(A):={\mathfrak{m} \subset A \mid \mathfrak{m} \text { is a maximal ideal }}$$
is called the maximal spectrum of $A$. For $X=\operatorname{Spec}(A)$ and $I \subset A$ an ideal
$$V(I):={P \in X \mid P \supset I}$$
is called the zero-set of $I$ in $X$. Note that $V(I)=\operatorname{supp}(A / I)$.

数学代写|交换代数代写Commutative Algebra代考|Projective Varieties

Affine varieties are the most important varieties as they are the building blocks for arbitrary varieties. Arbitrary varieties can be covered by open subsets which are affine varieties together with certain glueing conditions.

In modern treatments this glueing condition is usually coded in the notion of a sheaf, the structure sheaf of the variety. We are not going to introduce arbitrary varieties, since this would take us too deep into technical geometric constructions and too far away from commutative algebra.

However, there is one class of varieties which is the most important class of varieties after affine varieties and almost as closely related to algebra as affine ones. This is the class of the projective varieties.

What is the difference between affine and projective varieties? Affine varieties, for example $\mathbb{C}^n$, are in a sense open; travelling as far as we want, we can imagine the horizon – but we shall never reach infinity. On the other hand, projective varieties are closed (in the sense of compact, without boundary); indeed, we close up $\mathbb{C}^n$ by adding a “hyperplane at infinity” and, in this way, we domesticate infinity. The hyperplane at infinity can then be covered by finitely many affine varieties. In this way, finally, we obtain a variety covered by finitely many affine varieties, and we feel pretty well at home, at least locally.

However, the importance of projective varieties does not result from the fact that they can be covered by affine varieties, this holds for any variety. The important property of projective varieties is that they are closed, hence there is no escape to infinity. The simplest example demonstrating this are two parallel lines which do not meet in $\mathbb{C}^2$ but do meet in the projective plane $\mathbb{P}^2(\mathbb{C})$. This is what a perspective picture suggests, two parallel lines meeting at infinity.

数学代写|交换代数代写Commutative Algebra代考|Spectrum and Affine Schemes

A.3.1定义让$A$成为一个戒指。然后
$$\operatorname{Spec}(A):={P \subset A \mid P \text { is a prime ideal }}$$

$$\operatorname{Max}(A):={\mathfrak{m} \subset A \mid \mathfrak{m} \text { is a maximal ideal }}$$

$$V(I):={P \in X \mid P \supset I}$$

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。