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# 数学代写|交换代数代写Commutative Algebra代考|Filtrations and the Lemma of Artin-Rees

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## 数学代写|交换代数代写Commutative Algebra代考|Filtrations and the Lemma of Artin-Rees

In the next section, we shall study modules over local rings and the filtration induced by the powers of the maximal ideal. In this section, we provide the technical prerequisites.
Let $A$ be a Noetherian ring and $Q \subset A$ be an ideal.
Definition 5.4.1. A set $\left{M_n\right}_{n \geq 0}$ of submodules of an $A$-module $M$ is called $Q-$ filtration of $M$ if
(1) $M=M_0 \supset M_1 \supset M_2 \supset \ldots$
(2) $Q M_n \subset M_{n+1}$ for all $n \geq 0$.
A $Q$-filtration $\left{M_n\right}_{n \geq 0}$ of $M$ is called stable if $Q M_n=M_{n+1}$ for all sufficiently large $n$.

Example 5.4.2. Let $M$ be an $A$-module and $M_n:=Q^n M$ for $n \geq 0$. Then $\left{M_n\right}_{n \geq 0}$ is a stable $Q$-filtration of $M$.

Lemma 5.4.3. Let $\left{M_n\right}_{n \geq 0}$ and $\left{N_n\right}_{n \geq 0}$ be two stable $Q-$ filtrations of $M$. Then there exists some non-negative integer $n_0$ such that $M_{n+n_0} \subset N_n$ and $N_{n+n_0} \subset M_n$ for all $n \geq 0$.

Proof. We may assume that one of the two filtrations is the one of Example 5.4.2, say $N_n:=Q^n M$. Now, $\left{M_n\right}_{n \geq 0}$ being stable implies that there exists some non-negative integer $n_0$ such that $M_{n_0+n}=Q^n M_{n_0}$ for all $n \geq 0$. On the other hand, $Q^n M_{n_0} \subset Q^n M=N_n$ implies $M_{n_0+n} \subset N_n$.

As $\left{M_n\right}_{n \geq 0}$ is a $Q$-filtration, we have $Q M_n \subset M_{n+1}$ for all $n \geq 0$, which implies, in particular, $N_n=Q^n M=Q^n M_0 \subset M_n$. But $N_{n_0+n} \subset N_n$ which proves the claim.

## 数学代写|交换代数代写Commutative Algebra代考|The Hilbert–Samuel Function

The Hilbert-Samuel function is the counterpart to the Hilbert function in the local case. To a module $M$ over the local ring $(A, \mathfrak{m})$ and to an integer $n$, it associates the dimension of $M / \mathfrak{m}^n M$. Similarly to the homogeneous case, this function is a polynomial for large $n$, the Hilbert-Samuel polynomial. By passing to the associated graded module $\operatorname{Gr}{\mathfrak{m}}(M)=\bigoplus{\nu=0}^{\infty} \mathfrak{m}^\nu M / \mathfrak{m}^{\nu+1} M$, the results from the homogeneous case can be used.

Let $A$ be a local Noetherian ring with maximal ideal $m$. We assume (just for simplicity) that $K=A / \mathfrak{m} \subset A$. Moreover, let $Q$ be an $\mathfrak{m}$-primary ideal and $M$ a finitely generated $A$-module. Recall that the associated graded ring to $Q \subset A$ is defined as $\operatorname{Gr}Q(A)=\bigoplus{\nu=0}^{\infty}\left(Q^\nu / Q^{\nu+1}\right)$ (cf. Example 2.2.3).
Lemma 5.5.1. Let $\left{M_n\right}_{n \geq 0}$ be a stable Q-filtration of $M$, and let
$$\operatorname{HS}{\left{M_n\right}{n \geq 0}}(k):=\operatorname{dim}K\left(M / M_k\right) .$$ Moreover, suppose that $Q$ is generated by $r$ elements. Then (1) $\operatorname{HS}{\left{M_n\right}_{n \geq 0}}(k)<\infty$ for all $k \geq 0$;
(2) there exists a polynomial $\mathrm{HSP}{\left{M_n\right}{n \geq 0}}(t) \in \mathbb{Q}[t]$ of degree at most $r$ such that $\operatorname{HS}{\left{M_n\right}{n \geq 0}}(k)=\operatorname{HSP}{\left{M_n\right}{n \geq 0}}(k)$ for all sufficiently large $k$;
(3) the degree of $\mathrm{HSP}{\left{M_n\right}{n \geq 0}}$ and its leading coefficient do not depend on the choice of the stable $Q$-filtration $\left{M_n\right}_{n \geq 0}$.

Proof. $\operatorname{Gr}Q(A)=\bigoplus{\nu \geq 0} Q^\nu / Q^{\nu+1}$ is a graded $K$-algebra (Example 2.2.3), which is generated by $r$ elements of degree 1 . Now, let
$$\operatorname{Gr}{\left{M_n\right}}(M):=\bigoplus{\nu \geq 0} M_\nu / M_{\nu+1}$$

## 数学代写|交换代数代写Commutative Algebra代考|Filtrations and the Lemma of Artin-Rees

5.4.1.定义一个$A$ -模块$M$的一组$\left{M_n\right}{n \geq 0}$子模块称为$M$ if的$Q-$过滤 (1) $M=M_0 \supset M_1 \supset M_2 \supset \ldots$ (2) $Q M_n \subset M{n+1}$为所有$n \geq 0$。

## 数学代写|交换代数代写Commutative Algebra代考|The Hilbert–Samuel Function

Hilbert- samuel函数是局部情况下Hilbert函数的对应物。对于本地环$(A, \mathfrak{m})$上的模块$M$和整数$n$，它关联了$M / \mathfrak{m}^n M$的维度。类似于齐次情况，这个函数是一个多项式对于$n$大，Hilbert-Samuel多项式。通过传递给相关的分级模块$\operatorname{Gr}{\mathfrak{m}}(M)=\bigoplus{\nu=0}^{\infty} \mathfrak{m}^\nu M / \mathfrak{m}^{\nu+1} M$，可以使用齐次情况的结果。

(2)存在多项式 $\mathrm{HSP}{\left{M_n\right}{n \geq 0}}(t) \in \mathbb{Q}[t]$ 最多的程度 $r$ 这样 $\operatorname{HS}{\left{M_n\right}{n \geq 0}}(k)=\operatorname{HSP}{\left{M_n\right}{n \geq 0}}(k)$ 对于所有足够大的 $k$；
(3)的程度 $\mathrm{HSP}{\left{M_n\right}{n \geq 0}}$ 且其领先系数不依赖于稳态的选择 $Q$-过滤 $\left{M_n\right}_{n \geq 0}$．

$$\operatorname{Gr}{\left{M_n\right}}(M):=\bigoplus{\nu \geq 0} M_\nu / M_{\nu+1}$$

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