Posted on Categories:Commutative Algebra, 交换代数, 数学代写

# 数学代写|交换代数代写Commutative Algebra代考|MATH483 The homological tool for Noetherian rings

avatest™

## 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|交换代数代写Commutative Algebra代考|The homological tool for Noetherian rings

The purpose of this section is to develop enough of the basic material needed for the homological approach to regular local rings. Fatally, by so doing one is led to introducing sufficiently many details to make the section self-contained. The theme has been treated in many excellent sources, starting with the celebrated book by H. Cartan and S. Eilenberg. By going through the section, the reader will recognize the original work of the classical sources, and made explicit as much as possible.
6.2.1 Projective modules
The basic notion is encapsulated in the following conditions.
Proposition 6.2.1. Let $R$ denote a ring and let $M$ stand for an R-module. The following conditions are equivalent:
(i) (Lifting) Given $R$-modules $N, L$, a homorphism $\varphi: M \rightarrow L$ and a surjective homomorphism $\psi: N \rightarrow L$, there is a homomorphism $\chi: M \rightarrow N$ such that $\varphi=\psi \circ \chi$; in other words, for any surjective homomorphism $\psi: N \rightarrow L$ the induced homomorphism $\operatorname{Hom}_R(M, N) \rightarrow \operatorname{Hom}_R(M, L)$ is also surjective.
(ii) (Splitting) Every surjective homomorphism $\varphi: N \rightarrow M$ splits, i.e., there exists a homomorphism backwards $\psi: M \rightarrow N$ such that the composite $\Phi \circ \psi$ is the identity map of $M$.
(iii) (Direct summand) $M$ is a direct summand of a free $R$-module.

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

Corollary 6.2.6. Let $R$ denote a Noetherian ring and let $M$ stand for a finitely generated $R$-module. Then $M$ is projective if and only if $M_{\wp}$ is $R_p$-free for every $\wp \in \operatorname{Spec} R$.

Proof. The “only if” direction follows from Proposition 6.2.3 (and holds true without any hypothesis on either $R$ or $M$, by the previous contents).

Alas, simple as it sounds, the converse statement has no known proof that does not essentially use some functorial argument. The result is a consequence of the more general statement that a short exact sequence $0 \rightarrow L \rightarrow N \rightarrow M \rightarrow 0$ of finitely generated modules over a Noetherian ring $R$ splits if and only if it splits locally everywhere. One applies the functor $\operatorname{Hom}_R\left(M, _\right)$to this sequence yielding a left exact sequence
$$0 \rightarrow \operatorname{Hom}_R(M, L) \rightarrow \operatorname{Hom}_R(M, N) \rightarrow \operatorname{Hom}_R(M, M)$$
By assumption, the last sequence splits locally everywhere since Hom(, ) commutes with localizations under the present finiteness hypotheses; in particular, the rightjective because the cokernel vanishes locally everywhere (hence is zero). Now, any element of $\operatorname{Hom}_R(M, N)$ in the preimage of the identity map of $M$ is a splitting map for the original exact sequence.

## 数学代写|交换代数代写Commutative Algebra代考|The homological tool for Noetherian rings

6.2.1 投影模块

(i) (Lifting) Given $R$-模块 $N, L$,一个同态 $\varphi: M \rightarrow L$ 和满射同态 $\psi: N \rightarrow L$, 存在同态 $\chi: M \rightarrow N$ 这样 的。
(ii) (分裂) 每个满射同态 $\varphi: N \rightarrow M$ 分裂，即向后存在同态 $\psi: M \rightarrow N$ 这样复合 $\Phi \circ \psi$ 是身份映射 $M . . . .$.
(iii) (直接要求) $M$ 是一个自由的直接求和 $R$-模块。

## 数学代号|交换代数代与Commutative Algebra代㛈|Corollary 都免费 $\wp \in \operatorname{Spec} R$.

$$0 \rightarrow \operatorname{Hom}_R(M, L) \rightarrow \operatorname{Hom}_R(M, N) \rightarrow \operatorname{Hom}_R(M, M)$$

## 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。