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# 数学代写|交换代数代写Commutative Algebra代考|The graded Hilbert function

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## 数学代写|交换代数代写Commutative Algebra代考|The graded Hilbert function

This is the case where one aims at an enlargement of the setup in Section 2.7. Namely, one needs the extended notions of graded structures, as discussed in the beginning of Chapter 7.
The basic sine qua non result is the following elementary observation.
Lemma 7.4.5. Let $A$ stand for a commutative ring and let $R$ denote an $\mathbb{N}$-graded finitely generated A-algebra, with $R_0=A$. Let $M$ denote a finitely generated $\mathbb{Z}$-graded R-module. Then every homogeneous part of $M$ is a finitely generated A-module.

Proof. Since $R$ is graded, one can assume that it is generated by a set $\left{f_1, \ldots, f_r\right}$ of homogeneous elements of degrees, say, $d_1 \leq \cdots \leq d_r$. Consider the $\mathbb{N}$-graded polynomial ring $A\left[x_1, \ldots, x_r\right]$, where $\operatorname{deg}\left(x_i\right)=d_i$. Consider the surjective homomorphism of $A$-algebras $A\left[x_1, \ldots, x_r\right] \rightarrow R$ such that $x_i \mapsto f_i$, for $i=1, \ldots, r$. Since this way the grading of $R$ is induced by that of $A\left[x_1, \ldots, x_r\right]$, one may assume that $R=A\left[x_1, \ldots, x_r\right]$. In this case, the result follows by an obvious reasoning from the analogous one for a standard polynomial ring.

This takes care of the case where $M=R$. For the general case, $M$ is the image of a graded $R$-module homomorphism with source a direct sum of finitely many copies of $R$ or $R$ shifted by some degree. Therefore, the result follows from the previous case.
In particular, if the ground ring $A$ is Artinian, every homogeneous part of $M$ has finite length. This is the base for the following concept.

## 数学代写|交换代数代写Commutative Algebra代考|Selecta

The formula of van der Waerden (Theorem 2.7.25) for the multiplicity of cyclic modules over a polynomial ring now fully extends to the present general situation, where it is called perhaps improperly the (graded) associativity formula. One goes back to the general setup of Definition 7.4.6.
Proposition 7.4.13. Let $M$ denote a finitely generated graded R-module. Then
$$e(M)=\sum_{\wp} \lambda\left(M_{\wp}\right) e(R / \wp)$$
where $p$ runs through the minimal primes of $M$ of maximal dimension.
Proof. The basic strategy comes from the following additivity behavior of the multiplicity along an exact sequence.

Claim. Let $0 \rightarrow N^{\prime} \rightarrow N \rightarrow N^{\prime \prime} \rightarrow 0$ stand for a homogeneous exact sequence of finitely generated graded $R$-modules. Then
$$e(N)= \begin{cases}e\left(N^{\prime}\right)+e\left(N^{\prime \prime}\right) & \text { if } \operatorname{dim} N=\operatorname{dim} N^{\prime}=\operatorname{dim} N^{\prime \prime} \ e\left(N^{\prime}\right) & \text { if } \operatorname{dim} N^{\prime \prime}<\operatorname{dim} N \ e\left(N^{\prime \prime}\right) & \text { if } \operatorname{dim} N^{\prime}<\operatorname{dim} N\end{cases}$$

To see the claim, recall that the Hilbert function is additive on short exact sequences as the one given. Taking $n \gg 0$, one gets an exact sequence of the values of the respective Hilbert polynomials. Since their degrees are the respective dimensions of the terms, the appended leading coefficients are either incomparable or equal. Since, according to Proposition 5.2.6(vii), the only alternatives for the dimensions along the exact sequence are the ones stated in the claim, the alternatives for the respective multiplicities are as stated.

## 数学代写|交换代数代写Commutative Algebra代考|Selecta

$$e(M)=\sum_{\wp} \lambda\left(M_{\wp}\right) e(R / \wp)$$

$e(N)=\left{e\left(N^{\prime}\right)+e\left(N^{\prime \prime}\right) \quad\right.$ if $\operatorname{dim} N=\operatorname{dim} N^{\prime}=\operatorname{dim} N^{\prime \prime} e\left(N^{\prime}\right) \quad$ if $\operatorname{dim} N^{\prime \prime}<\operatorname{dim} N e\left(N^{\prime \prime}\right) \quad$ if $\operatorname{dim} N^{\prime}<\operatorname{dim}$

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