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# 数学代写|交换代数代写Commutative Algebra代考|Finite and Integral Extensions

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## 数学代写|交换代数代写Commutative Algebra代考|Finite and Integral Extensions

This section contains the basic algebraic theory of finite and algebraic extensions and their relationship. Moreover, important criteria for integral dependence (Proposition 3.1.3) and finiteness (Proposition 3.1.5) are proven.
Definition 3.1.1. Let $A \subset B$ be rings.
(1) $b \in B$ is called integral over $A$ if there is a monic polynomial $f \in A[x]$ satisfying $f(b)=0$, that is, $b$ satisfies a relation of degree $p$,
$$b^p+a_1 b^{p-1}+\cdots+a_p=0, \quad a_i \in A$$
for some $p>0$.
(2) $B$ is called integral over $A$ or an integral extension of $A$ if every $b \in B$ is integral over $A$.
(3) $B$ is called a finite extension of $A$ if $B$ is a finitely generated $A$-module.
(4) If $\varphi: A \rightarrow B$ is a ring map then $\varphi$ is called an integral, respectively finite, extension if this holds for the subring $\varphi(A) \subset B$.

If there is no doubt about $\varphi$, we say also, in this situation, that $B$ is integral, respectively finite, over $A$. Often we omit $\varphi$ in the notation, for example we write $I M$ instead of $\varphi(I) M$ if $I \subset A$ is an ideal and $M$ a $B$-module.

## 数学代写|交换代数代写Commutative Algebra代考|The Integral Closure

We explain the notion of integral closure by an example. Assume we have a parametrization of an affine plane curve which is given by a polynomial $\operatorname{map} \mathbb{A}^1 \rightarrow \mathbb{A}^2, t \mapsto(x(t), y(t))$ such that $t$ is contained in the quotient field $K(x(t), y(t))$ of $A=K[x(t), y(t)]$. Let $A \subset B=K[t]$ denote the corresponding ring map, then $t$ is integral over $A$ and $A[t]=B$. We shall see that $K[t]$ is integrally closed in the quotient field $Q(A)=Q(K[t])=K(t)$, and the “smallest ring” with this property containing $A$ (Exercise 3.6.5). For example, $K\left[t^2, t^3\right] \subset K[t]$ corresponds to the parametrization of the cuspidal cubic (cf. Figure 3.2).

For arbitrary reduced affine curves with coordinate ring $A=K[x] / I$ the normalization of $A$, that is, the integral closure of $A$ in $Q(A)$, is the affine ring of a “desingularization” of the curve. For higher dimensional varieties, the normalization of the coordinate ring will not necessarily be a desingularization, but an improvement of the singularities, for example, the codimension of the singular locus will be $\geq 2$. Here we shall treat only some algebraic properties of the normalization.

More generally, we shall study the process associating to a ring extension $A \subset B$ the smallest subring $\widetilde{A} \subset B$ containing all elements of $B$ which are integral over $A$.

Definition 3.2.1. Let $A \subset B$ be a ring extension and $I \subset A$ be an ideal (the case $I=A$ is not excluded). An element $b \in B$ which satisfies a relation
$$b^n+a_1 b^{n-1}+\cdots+a_n=0, \quad a_i \in I$$
is called integral over $I$. We denote by
$$C(I, B)={b \in B \mid b \text { integral over } I}$$
the (weak) integral closure of $I$ in $B$. If, moreover, $a_i \in I^i$, we say that $b$ is strongly integral over $I$ and call
$$C_s(I, B)={b \in B \mid b \text { strongly integral over } I}$$
the strong integral closure of $I$ in $B$.

## 数学代写|交换代数代写Commutative Algebra代考|Finite and Integral Extensions

3.1.1.定义让$A \subset B$响起来吧。
(1)若存在一个单多项式$f \in A[x]$满足$f(b)=0$，即$b$满足次关系$p$，则称$b \in B$为对$A$的积分;
$$b^p+a_1 b^{p-1}+\cdots+a_p=0, \quad a_i \in A$$

(2) $B$称为$A$上的积分，如果每个$b \in B$都是$A$上的积分，则称为$A$的积分扩展。
(3)如果$B$是一个有限生成的$A$ -模块，则$B$称为$A$的有限扩展。
(4)如果$\varphi: A \rightarrow B$是一个环映射，那么$\varphi$被称为一个积分，分别是有限的扩展，如果这对子映射$\varphi(A) \subset B$成立。

## 数学代写|交换代数代写Commutative Algebra代考|The Integral Closure

3.2.1.定义设$A \subset B$为环扩展，$I \subset A$为理想($I=A$不排除这种情况)。满足关系的元素$b \in B$
$$b^n+a_1 b^{n-1}+\cdots+a_n=0, \quad a_i \in I$$

$$C(I, B)={b \in B \mid b \text { integral over } I}$$
$B$中$I$的(弱)积分闭包。此外，如果$a_i \in I^i$，我们说$b$是$I$的强积分，并称
$$C_s(I, B)={b \in B \mid b \text { strongly integral over } I}$$
$B$中$I$的强整闭包。

## MATLAB代写

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