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# 数学代写|交换代数代写COMMUTATIVE ALGEBRA代写|MATH4312 Hilbert characteristic function

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## 数学代写|交换代数代写COMMUTATIVE ALGEBRA代写|Basics on the underlying graded structures

A more comprehensive treatment of graded structures will be considered in Chapter $7 .$ Here, one focus on the following special setup: $R:=k\left[x_{0}, \ldots, x_{n}\right]$ stands for a polyno-mial ring over a field $k$. One endows $R$ with a structure of graded ring, by which one means the decomposition
$$R=\bigoplus_{t \geq 0} R_{t}, \quad R_{t}=k x_{0}^{t}+k x_{0}^{t-1} x_{1}+\cdots+k x_{n}^{t} \subset R .$$
The $k$-vector space $R_{t}$, spanned by the homogeneous polynomials of degree $t$, is called the $t$ th graded part of $R$. An ideal $I \subset R$ is homogeneous if it can be generated by homogeneous polynomials or, equivalently, if $I=\bigoplus_{t \geq 0} I_{t}$, where $I_{t}:=I \cap R_{t}$.

Often a homogeneous polynomial of degree $t$ will be called a $t$-form. Given a homogeneous ideal $I$, an important related degree is the initial degree of $I$, defined to be the least $t \geq 0$ such that $I_{t} \neq 0$.

Perhaps the first feature of homogeneous ideals is that the property of being prime or primary can be verified solely by using homogeneous test elements.

## 数学代写|交换代数代写COMMUTATIVE ALGEBRA代写|First results

Clearly, the residue ring $R / I$ inherits same sort of grading as $R$, where $(R / I){t}=R{t} / I_{t}$. Therefore, as $k$-vector spaces,
$$\operatorname{dim}{k}(R / I){t}=\operatorname{dim}{k} R{t}-\operatorname{dim}{k} I{t}=\left(\begin{array}{c} t+n \ n \end{array}\right)-\operatorname{dim}{k} I{t} .$$
Thus, the vector dimensions of $(R / I){t}$ and $I{t}$ differ by a known number, hence they are theoretically and computationally fairly interchangeable.

Definition 2.7.9. Let $I \subset R$ denote a homogeneous ideal. The Hilbert function of $R / I$ is the numerical function
$$H\left(R / I \text {,_ }^{\prime}\right): \mathbb{N} \longrightarrow \mathbb{N}, \quad t \mapsto \operatorname{dim}{k}(R / I){t} .$$
In particular, $H(R, t)=\left(\begin{array}{c}n+t \ t\end{array}\right)$ for every $t \geq 0$. This notion is attached to a graded ring, but it is customary to extend it to graded modules, which includes homogeneous ideals. For this reason, one frequently defines the Hilbert function of the ideal $I$ :
$$H(I, t):=\operatorname{dim}{k} I{t}=\left(\begin{array}{c} t+n \ n \end{array}\right)-H(R / I, t) .$$
Clearly, as already observed, the two are interchangeable.
Classically, side wise with the Hilbert function one also talks about the Hilbert series of $R / I$ as the generating function of $H\left(R / I\right.$,). It will be denoted $H{R / I}(t)$. Thus, $H_{R / I}(t)=\sum_{i \geq 0} H(R / I, t) t^{i}$. It can be shown that it is a rational function in $t$, whose denominator is $(1-t)^{d}$, where $d=\operatorname{dim} R / I$. The numerator is a polynomial that can be written down as soon as one knows the minimal free resolution (Proposition 7.4.11) of $R / I$ over $R$. The Hilbert series will be considered in more detail in the general case of graded modules (Section 7.4.2).
An important feature of combinatorics is the fact that
$$\left(\begin{array}{c} t+n \ n \end{array}\right)=\frac{1}{n !} t^{n}-\frac{1}{n !}\left(\begin{array}{c} n+1 \ 2 \end{array}\right) t^{n-1}+\text { lower order terms in } t,$$
a polynomial expression in $t$ of order $n$ and with coefficient of the higher order term equal to $1 / n$ !. The following example throws further light on this matter.

## 数学代写|交换代数代写COMMUTATIVE ALGEBRA代写|Basics on the underlying graded structures

$$R=\bigoplus_{t \geq 0} R_{t}, \quad R_{t}=k x_{0}^{t}+k x_{0}^{t-1} x_{1}+\cdots+k x_{n}^{t} \subset R .$$

$\mathrm{~ 也 许 対 次 理 想 的 第 一 个 特 征 是 ， 可 以 仅 通 过 使 用 齐 次 测 试 元 靺 来 验 证 是 否 为 表}$ 性。

## 数学代写|交换代数代写 COMMUTATIVE ALGEBRA代写|First results

$$H(I, t):=\operatorname{dim} k I t=(t+n n)-H(R / I, t) .$$

(命题 7.4.11) 就可以写下来 $R / I$ 超过 $R$. 希尔伯特级数将在分级模块的一般情况下更详 细地考虑（第 7.4.2 节）。

$$(t+n n)=\frac{1}{n !} t^{n}-\frac{1}{n !}(n+12) t^{n-1}+\text { lower order terms in } t,$$

## MATLAB代写

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