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

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## 数学代写|交换代数代写commutative algebra代考|Local rings and symbolic powers

If $P \subset R$ is a prime ideal and $\mathfrak{S}=R \backslash P$, then the ring $\mathfrak{S}^{-1} R$ contains a unique maximal ideal, namely $\varsigma^{-1} P$.

Denote $\mathcal{S}^{-1} R=R_{P}$, calling it the local ring of $R$ at $P$. Similarly, given an ideal $I \subset R$, set $\mathfrak{S}^{-1} I=I_{P}$. In this notation, the unique maximal ideal of $R_{P}$ is $P_{P}$ and, in particular, the prime ideals of $R_{P}$ correspond bijectively to those of $R$ contained in $P$. The passage from $R$ to $R_{P}$ via the natural homomorphism $t: R \longrightarrow R_{P}$ is called localization at $P$. (The newcomer is recommended not to use this terminology for other rings of fractions.)
The field $R_{P} / P_{P}$ is called the residue field of $P$ and hids a IIdjur role in the theury. Taking $\mathfrak{T}=R / P-{\overline{0}}$, this field is isomorphic to $\widetilde{T}^{-1}(R / P)$, the field of fractions of $R / P$.
Motivated by this, one introduces the following terminology.
Definition 2.1.6. A ring is local if it has a unique maximal ideal.
Quite often such a ring is called quasi-local, while local is used in the case where $R$ is moreover Noetherian (next chapter). Here, no such distinction in terminology will be made. A more relaxed condition requires that the ring have only finitely many maximal ideals, in which case it is called semilocal. Often a property of a local ring can be extended to a semilocal ring.

One great advantage of working with a Noetherian local ring $R$ is that the notion of minimal number of generators of an ideal $I \subset R$ is well-defined in the sense that any set of generators with no superfluous elements has the same cardinality. Such a property is better understood in terms of passage to the associated $(R / \mathrm{m})$-vector space $I / \mathrm{m} I$, where $\mathfrak{m} \subset R$ denotes the unique maximal ideal of $R$. The main result in this regard is Lemma 2.5.24, which delivers the basic techniques to handle these rings.
One important application of localization at a prime ideal is given by the notion of symbolic powers (see Theorem $2.4 .9$ for its geometric impact). The definition is surprisingly simple.

## 数学代写|交换代数代写commutative algebra代考|Integral ring extensions

Let $R \subset S$ be a ring extension.
An element $b \in S$ is said to be integral over $R$ if it is a root of a MONIC polynomial $f(X) \in R[X]$. Equivalently, $b$ is integral over $R$ if the kernel of the $R$-algebra map $R[X] \rightarrow S$, such that $X \mapsto b$, contains a monic polynomial. If this is the case, the resulting relation obtained by substituting for $b$ is called an equation of integral dependence.
The following criterion of integrality opens the gates to the theory. One should note its similarity to a well-known test for algebraic elements in a field extension. To state it, one recurs to the notion of a module and of a set of generators (see Chapter 3 ). Although it may look abstruse to introduce this notion at this early point, think about the elegance and quickness it affords in the argument below.

Proposition 2.2.1. Let $R \subset S$ be a ring extension and let $b \in S$. The following conditions are equivalent:
(i) $b$ is integral over $R$.
(ii) The subring $R[b] \subset S$ is a finitely generated R-module.
(iii) $R[b]$ is contained in a subring $T \subset S$ which is a finitely generated R-module.
Proof. (i) $\Rightarrow$ (ii) Say, $b^{n}+a_{1} b^{n-1}+\cdots+a_{0}=0$, where $a_{i} \in R$. Clearly, then $b^{n} \in$ $\sum_{i=0}^{n-1} R b^{i}$, the latter meaning the $R$-linear combinations of the powers $1, b, \ldots, b^{n-1}$, i.e., the $R$-submodule generated by them. By recurrence, multiplying both members of the above equation of integral dependence by $b$ yields $b^{m} \in \sum_{i=0}^{n-1} R b^{i}$ for every $m \geq 0$. This gives $R[b]=\sum_{i=0}^{n-1} R b^{i}$, as stated.
(ii) $\Rightarrow$ (iii) Obvious.

## 数学代写|交换代数代写commutative algebra代考|The Cohen–Seidenberg theorems

Next is a fundamental property of integral extensions $R \subset S$ with respect to multiplicatively closed subsets $\mathfrak{\Im} \subset R$. This single theorem unifies all other related results, often proved separately ( $c f$. [36] for the main source).

Theorem 2.2.6 (Unified Cohen-Seidenberg theorem). Let $R \subset S$ be an integral extension, let $\mathfrak{\Im} \subset R$ be a multiplicatively closed subset and let $Q \subset S$ be a prime ideal not intersecting $\mathfrak{S}$. Then $Q \cap R$ does not intersect $\mathfrak{S}$ and the following conditions are equivalent:
(i) $Q$ is maximal among the ideals of $S$ not intersecting $\mathfrak{S}$.
(ii) $Q \cap R$ is maximal among the ideals of $R$ not intersecting $\mathfrak{S}$.
Proof. Clearly, $Q \cap R$ does not intersect $\mathfrak{S}$ since $(Q \cap R) \cap \mathfrak{S}=Q \cap \mathfrak{S}$.
(i) $\Rightarrow$ (ii) Assuming the contrary, let $Q \cap R \varsubsetneqq I$, where $I \subset R$ is an ideal not intersecting $\mathfrak{S}$. Say, $a \in I \backslash(Q \cap R)$. Clearly, $a \notin Q$, so $Q \subset(Q, a)$ is a proper inclusion, hence $(Q, a) \cap \mathfrak{S} \neq \emptyset$ by assumption. Thus, let $\mathfrak{s} \in(Q, a) \cap \mathfrak{S}$, say, $\mathfrak{s}=q+a b$, with $q \in Q$ and $b \in S$. Since $R \subset S$ is integral, there is an equation of integral dependence for $b$ over $R$
$$b^{n}+a_{1} b^{n-1}+\cdots+a_{n-1} b+a_{n}=0, \quad a_{i} \in R$$ Multiplying out by $a^{n}$, yields an equation of integral dependence for $a b$ over $R$. Taking in account the form of $s$, one can see that the element $c=: \mathfrak{s}^{n}+\left(a_{1} a\right) \mathfrak{s}^{n-1}+\cdots+a_{n} a^{n}$ belongs to $Q$ and, clearly, to $R$, hence $c \in Q \cap R \subset I$. On the other hand, $c$ is of the form $\mathfrak{s}^{n}+a^{\prime} a$, for a suitable $a^{\prime} \in R$. Since $a \in I$ to start with, then $\mathfrak{s}^{n} \in I$. Since $\mathfrak{S}$ is multiplicatively closed, then $\mathfrak{s}^{n} \in S$. Therefore, $\mathfrak{s}^{n} \in I \cap \mathfrak{S}$, thus contradicting the assumption $I \cap \mathfrak{S}=\emptyset$.

## 数学代写|交换代数代写commutative algebra代考|Local rings and symbolic powers

）R磷/磷磷称为残差场磷并在理论中隐藏了 IIdjur 的角色。服用吨=R/磷−0¯, 这个域同构于吨~−1(R/磷), 分数域R/磷.

## 数学代写|交换代数代写commutative algebra代考|Integral ring extensions

(i)b是积分超过R.
(ii) 子环R[b]⊂小号是一个有限生成的 R 模。
㈢R[b]包含在子环中吨⊂小号这是一个有限生成的 R 模。

(二)⇒(iii) 显而易见。

## 数学代写|交换代数代写commutative algebra代考|The Cohen–Seidenberg theorems

(i)问在理想中是最大的小号不相交小号.
(二)问∩R在理想中是最大的R不相交小号.

（一世）⇒(ii) 假设相反，让问∩R⫋我， 在哪里我⊂R是不相交的理想小号. 说，一个∈我∖(问∩R). 清楚地，一个∉问， 所以问⊂(问,一个)是一个适当的包含，因此(问,一个)∩小号≠∅通过假设。因此，让s∈(问,一个)∩小号， 说，s=q+一个b， 和q∈问和b∈小号. 自从R⊂小号是积分，有一个积分依赖方程b超过R

bn+一个1bn−1+⋯+一个n−1b+一个n=0,一个一世∈R乘以一个n, 产生一个积分依赖方程一个b超过R. 考虑到形式s, 可以看出元素C=:sn+(一个1一个)sn−1+⋯+一个n一个n属于问并且，显然，R， 因此C∈问∩R⊂我. 另一方面，C是形式sn+一个′一个, 为一个合适的一个′∈R. 自从一个∈我开始，然后sn∈我. 自从小号是乘法闭的，那么sn∈小号. 所以，sn∈我∩小号，因此与假设相矛盾我∩小号=∅.

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