### 数学代写|交换代数代写commutative algebra代考|MTH2121

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• 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

## 数学代写|交换代数代写commutative algebra代考|Some Facts Concerning Quotients and Localizations

Let us begin by recalling the following result on quotients. Let $a$ be an ideal of a ring $\mathbf{A}$. When needed, the canonical mapping will be denoted by $\pi_{\mathbf{A}, \mathfrak{a}}: \mathbf{A} \rightarrow \mathbf{A} / \mathfrak{a}$.
The quotient ring $\left(\mathbf{A} / \mathfrak{a}, \pi_{\mathbf{A}, \mathfrak{a}}\right)$ is characterized, up to unique isomorphism, by the following universal property.
1.1 Fact (Characteristic property of the quotient by the ideal a) A ring homomorphism $\psi: \mathbf{A} \rightarrow \mathbf{B}$ is factorized by $\pi_{\mathbf{A}, \mathfrak{a}}$ if and only if $\mathfrak{a} \subseteq \operatorname{Ker} \psi$, meaning $\psi(\mathfrak{a}) \subseteq\left{0_{\mathrm{B}}\right}$. In this case, the factorization is unique.

Explanation regarding the figure. In a figure of the type found above, everything but the morphism $\theta$ corresponding to the dotted arrow is given. The exclamation mark signifies that $\theta$ makes the diagram commute and that it is the unique morphism with this property.

We denote by $M /$ a $M$ the $\mathbf{A} / \mathfrak{a}$-module obtained from the quotient of the A-module $M$ by the submodule generated by the elements $a x$ for $a \in \mathfrak{a}$ and $x \in M$. This module can thus be defined through the extension of scalars to $\mathbf{A} / \boldsymbol{a}$ from the $\mathbf{A}$-module $M$ (see p. 191, and Exercise IV-5).

Let us move on to localizations, which are very analogous to quotients (we will return to this analogy in further detail on p. 635). In this work, when referring to a monoid contained within a ring (i.e. a submonoid of a ring) we always assume a subset of the ring which contains 1 and is closed under multiplication.

For a given ring $\mathbf{A}$, we denote by $\mathbf{A}^{\times}$the multiplicative group of invertible elements, also called the group of units.

If $S$ is a monoid, we denote by $\mathbf{A}{S}$ or $S^{-1} \mathbf{A}$ the localization of $\mathbf{A}$ at $S$. Every element of $\mathbf{A}{S}$ can be written in the form $x / s$ with $x \in \mathbf{A}$ and $s \in S$.

By definition we have $x_{1} / s_{1}=x_{2} / s_{2}$ if there exists an $s \in S$ such that $s s_{2} x_{1}=$ $s s_{1} x_{2}$. When needed, we will denote by $j_{\mathbf{A}, S}: \mathbf{A} \rightarrow \mathbf{A}_{S}$ the canonical mapping $x \mapsto x / 1$.

The localized ring ( $\left.\mathbf{A}{S}, j{\mathbf{A}, S}\right)$ is characterized, up to unique isomorphism, by the following universal property.

## 数学代写|交换代数代写commutative algebra代考|The Basic Local-Global Principle

We will study the general workings of the local-global principle in commutative algebra in Chap. XV. However, we will encounter it at every turn, under different forms adapted to each situation. In this section, an essential instance of this principle is given as it is so simple and efficient that it would be a pity to go without it any longer.

The local-global principle affirms that certain properties are true if and only if they are true after “sufficiently many” localizations. In classical mathematics we often invoke localization at every maximal ideal. It is a lot of work and seems a bit mysterious, especially from an algorithmic point of view. We will use simpler (and less intimidating) versions in which only a finite number of localizations are used.

Here is a characterization from classical mathematics.
2.2 Fact* Let $S_{1}, \ldots, S_{n}$ be monoids in a nontrivial ring A (i.e., $1 \neq_{\mathbf{A}} 0$ ). The monoids $S_{i}$ are comaximal if and only if for every prime ideal (resp. for every maximal ideal) $\mathfrak{p}$ one of the $S_{i}$ is contained within $\mathbf{A} \backslash \mathfrak{p}$.

D Let $\mathfrak{p}$ be a prime ideal. If none of the $S_{i}$ ‘s are contained in $\mathbf{A} \backslash \mathfrak{p}$ then for each $i$ there exists some $s_{i} \in S_{i} \cap \mathfrak{p}$. Consequently, $s_{1}, \ldots, s_{n}$ are not comaximal.

Conversely, suppose that for every maximal ideal $\mathfrak{m}$ one of the $S_{i}$ ‘s is contained within $\mathbf{A} \backslash \mathfrak{m}$ and let $s_{1} \in S_{1}, \ldots, s_{n} \in S_{n}$ then the ideal $\left\langle s_{1}, \ldots, s_{n}\right\rangle$ is not contained in any maximal ideal. Thus it contains $1 .$

We denote by $\mathbf{A}^{m \times p}$ or $\mathbb{M}{m, p}(\mathbf{A})$ the $\mathbf{A}$-module of $m$-by- $p$ matrices with coefficients in $\mathbf{A}$, and $\mathbb{M}{n}(\mathbf{A})$ means $\mathbb{M}{n, n}(\mathbf{A})$. The group of invertible matrices is denoted by $\mathbb{G L}{n}(\boldsymbol{\Lambda})$, the subgroup consisting of the matrices of determinant 1 is denoted by $\mathbb{S L}{n}(\mathbf{A})$. The subset of $\mathbb{M}{n}(\mathbf{A})$ consisting of the projection matrices (i.e. matrices $F$ such that $F^{2}=F$ ) is denoted by $\mathbb{A}_{n}(\mathbf{A})$. The acronyms are explained as follows: $\mathbb{G L}$ for linear group, $\mathbb{S L}$ for special linear group and $\mathbb{A} G$ for affine Grassmannian.

## 数学代写|交换代数代写commutative algebra代考|Some Facts Concerning Quotients and Localizations

$1.1$ 事实 (理想商的特征性质 $a$ ) 环同态 $\psi: \mathbf{A} \rightarrow \mathbf{B}$ 被分解为 $\pi_{\mathbf{A}, \mathrm{a}}$ 当且仅当 $\mathfrak{a} \subseteq \operatorname{Ker} \psi$ ，意义 Ipsi(Imathfrak{a}) Isubseteq lleft{0_(Imathrm{B}}}right} . 在这种情况下，分解是唯一的。

## 数学代写|交换代数代写commutative algebra代考|The Basic Local-Global Principle

$2.2$ 事实*让 $S_{1}, \ldots, S_{n}$ 是非平凡环 A 中的么半群 (即， $1 \neq \mathbf{A}{\mathbf{A}} 0$ ) 。类半群 $S{i}$ 当且仅当对于每个素理想 (分别对 于每个最大理想) 都是共极大的 $\mathfrak{p}$ 中的一个 $S_{i}$ 包含在 $\mathbf{A} \backslash \mathfrak{p}$.
$\mathrm{D}$ 让p成为一个首要的理想。如果没有 $S_{i}$ 的包含在 $\mathbf{A} \backslash \mathfrak{p}$ 然后对于每个 $i$ 存在一些 $s_{i} \in S_{i} \cap \mathfrak{p}$. 最后， $s_{1}, \ldots, s_{n}$ 不 是共大的。

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