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# 数学代写|交换代数代写Commutative Algebra代考|MATH483 Homological methods

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## 数学代写|交换代数代写Commutative Algebra代考|Homological methods

While large sectors of commutative algebra have their origins in or are closely related to the geometry of algebraic varieties, there is one sector whose development was quite independent, and that is the homological tool. This has been organized and unified in the book of Cartan-Eilenberg, which became the state-of-the-art of the subject. The spectacular impact to commutative algebra followed suit in the work of M. Auslander, D. Buchsbaum, D. Northcott, D. Rees and J. P. Serre, to mention the first mentors.
6.1 Regular local rings
Although the terminology “regular local ring” has a deep entrenchment in the theory of smooth algebraic varieties and in homology in the 1950s, it is surprising how one can travel quite a bit in shallower waters to grasp its preliminaries. The purpose of the first part is to explore as much as possible the natural position of this concept amidst the basic notions of commutative ring theory.

## 数学代写|交换代数代写Commutative Algebra代考|Relation to basic invariants

For the sake of the present discussion, $(R, \mathfrak{m})$ is a Noetherian local ring. Recall that $\mu(I)$ denotes the minimal number of generators of an ideal $I \subset \mathfrak{m}$. One knows that $\operatorname{depth} R \leq \operatorname{dim} R \leq \mu(\mathfrak{m})$, where the rightmost inequality follows from Krull’s prime ideal theorem.

The leftmost inequality being an equality means that $R$ is Cohen-Macaulay. On the other hand, equality throughout means that $m$ can be generated by an $R$-sequence. Indeed, far more generally, one can show: let $R$ be a Noetherian ring, $M$ a finitely generated $R$-module and $I \subset R$ an ideal generated by $n$ elements such that $I M \neq M$. If $\operatorname{depth}I M=n$, then $I$ can be generated by an $M$-sequence of length $n$. For the proof, one applies iteratively prime avoidance (Lemma 2.5.22). Say, $I=\left(a_1, \ldots, a_n\right)$. Then one gets a set of generators in triangular shape as follows: \begin{aligned} & a_1+\lambda{1,2} a_2+\cdots+\lambda_{1, n} a_n=a_1+u_1 \ & a_2+\lambda_{2,3} a_3+\cdots+\lambda_{2, n} a_n=a_2+u_2 \end{aligned}

## 数学代写|交换代数代写Commutative Algebra代考|Homological methods

Buchsbaum、D. Northcott、D. Rees 和 JP Serre 的工作对交换代数产生了巨大的影响，这些都是第一位导 师。
6.1 常规本地环

## 数学代写|交换代数代写Commutative Algebra代考|Relation to basic invariants

$$a_1+\lambda 1,2 a_2+\cdots+\lambda_{1, n} a_n=a_1+u_1 \quad a_2+\lambda_{2,3} a_3+\cdots+\lambda_{2, n} a_n=a_2+u_2$$

## 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。