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# 数学代写|交换代数代写Commutative Algebra代考|Zero–dimensional Primary Decomposition

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## 数学代写|交换代数代写Commutative Algebra代考|Zero–dimensional Primary Decomposition

In this section we shall give an algorithm to compute a primary decomposition for zero-dimensional ideals in a polynomial ring over a field of characteristic 0. This algorithm was published by Gianni, Trager, and Zacharias ([90]). Let $K$ be a field of characteristic 0 . In the case of one variable $x$, any ideal $I \subset K[x]$ is a principal ideal and the primary decomposition is given by the factorization of a generator of $I$ : let $I=\langle f\rangle, f=f_1^{n_1} \ldots f_r^{n_r}$ with $f_i$ irreducible and $\left\langle f_i, f_j\right\rangle=K[x]$ for $i \neq j$, then $I=\left\langle f_1\right\rangle^{n_1} \cap \cdots \cap\left\langle f_r\right\rangle^{n_r}$ is the primary decomposition of $I$. In the case of $n$ variables, the univariate polynomial factorization is also an essential ingredient. We shall see that, after a generic coordinate change, the factorization of a polynomial in one variable leads to a primary decomposition. By definition, all associated prime ideals of a zero-dimensional ideal are maximal. We need the concept for an ideal in general position.

Definition 4.2.1.
(1) A maximal ideal $M \subset K\left[x_1, \ldots, x_n\right]$ is called in general position with respect to the lexicographical ordering with $x_1>\cdots>x_n$, if there exist $g_1, \ldots, g_n \in K\left[x_n\right]$ with $M=\left\langle x_1+g_1\left(x_n\right), \ldots, x_{n-1}+g_{n-1}\left(x_n\right), g_n\left(x_n\right)\right\rangle$.
(2) A zero-dimensional ideal $I \subset K\left[x_1, \ldots, x_n\right]$ is called in general position with respect to the lexicographical ordering with $x_1>\cdots>x_n$, if all associated primes $P_1, \ldots, P_k$ are in general position and if $P_i \cap K\left[x_n\right] \neq$ $P_j \cap K\left[x_n\right]$ for $i \neq j$.

## 数学代写|交换代数代写Commutative Algebra代考|Higher Dimensional Primary Decomposition

In this section we show how to reduce the primary decomposition of an arbitrary ideal in $K[x]$ to the zero-dimensional case. We use the following idea:
Let $K$ be a field and $I \subset K[x]$ an ideal. Let $u \subset x=\left{x_1, \ldots, x_n\right}$ be a maximal independent set with respect to the ideal $I$ (cf. Definition 3.5.3) then $\emptyset \subset x \backslash u$ is a maximal independent set with respect to $I K(u)[x \backslash u]$ and, therefore, $I K(u)[x \backslash u] \subset K(u)[x \backslash u]$ is a zero-dimensional ideal (Theorem 3.5.1 (6)). Now, let $Q_1 \cap \cdots \cap Q_s=I K(u)[x \backslash u]$ be an irredundant primary decomposition (which we can compute as we are in the zero-dimensional case), then also $I K(u)[x \backslash u] \cap K[x]=\left(Q_1 \cap K[x]\right) \cap \cdots \cap\left(Q_s \cap K[x]\right)$ is an irredundant primary decomposition. It turns out that $I K(u)[x \backslash u] \cap K[x]$ is equal to the saturation $I:\left\langle h^{\infty}\right\rangle=\bigcup_{m>0} I:\left\langle h^m\right\rangle$ for some $h \in K[u]$ which can be read from an appropriate Gröbner basis of $I K(u)[x \backslash u]$. Assume that $I:\left\langle h^{\infty}\right\rangle=I:\left\langle h^m\right\rangle$ for a suitable $m$ (the ring is Noetherian). Then, using Lemma 3.3.6, we have $I=\left(I:\left\langle h^m\right\rangle\right) \cap\left\langle I, h^m\right\rangle$. Because we computed already the primary decomposition for $I:\left\langle h^m\right\rangle$ (an equidimensional ideal of dimension $\operatorname{dim}(I))$ we can use induction, that is, apply the procedure again to $\left\langle I, h^m\right\rangle$.

This approach terminates because either $\operatorname{dim}\left(\left\langle I, h^m\right\rangle\right)<\operatorname{dim}(I)$ or the number of maximal independent sets with respect to $\left\langle I, h^m\right\rangle$ is smaller than the number of maximal independent sets with respect to $I$ (since $u$ is not an independent set with respect to $\left\langle I, h^m\right\rangle$ ). The basis of this reduction procedure to the zero-dimensional case is the following proposition:

Proposition 4.3.1. Let $I \subset K[x]$ be an ideal and $u \subset x=\left{x_1, \ldots, x_n\right}$ be a maximal independent set of variables with respect to $I$.
(1) $I K(u)[x \backslash u] \subset K(u)[x \backslash u]$ is a zero-dimensional ideal.
(2) Let $S=\left{g_1, \ldots, g_s\right} \subset I \subset K[x]$ be a Gröbner basis of $I K(u)[x \backslash u]$, and let $h:=\operatorname{lcm}\left(\mathrm{LC}\left(g_1\right), \ldots, \mathrm{LC}\left(g_s\right)\right) \in K[u]$, then
$$I K(u)[x \backslash u] \cap K[x]=I:\left\langle h^{\infty}\right\rangle,$$
and this ideal is equidimensional of dimension $\operatorname{dim}(I)$.
(3) Let $I K(u)[x \backslash u]=Q_1 \cap \cdots \cap Q_s$ be an irredundant primary decomposition, then also $I K(u)[x \backslash u] \cap K[x]=\left(Q_1 \cap K[x]\right) \cap \cdots \cap\left(Q_s \cap K[x]\right)$ is an irredundant primary decomposition.

## 数学代写|交换代数代写Commutative Algebra代考|Zero–dimensional Primary Decomposition

4.2.1.定义
(1)如果存在$g_1, \ldots, g_n \in K\left[x_n\right]$和$M=\left\langle x_1+g_1\left(x_n\right), \ldots, x_{n-1}+g_{n-1}\left(x_n\right), g_n\left(x_n\right)\right\rangle$，则在相对于$x_1>\cdots>x_n$的字典顺序的一般位置上称为极大理想$M \subset K\left[x_1, \ldots, x_n\right]$。

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