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# 数学代写|交换代数代写Commutative Algebra代考|MA8202 Sequences

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

While in the preceding examples, the given ideal had a kind of “external” character, in the sense that it carried a canonically defined set of generators, the present one will have some “internal” character, namely, will depend on admitting a set of generators of a particular behavior.

The following notion has been introduced by C. Huneke ([79]). It had previously been defined by M. Fiorentini ([57]) in a slightly different form.

Definition 7.3.25. A set of elements $a_1, \ldots, a_n$ in a ring $R$ is called a $d$-sequence if it satisfies the following equalities:
$$\left(a_{i_1}, \ldots, a_{i_l}\right): a_t a_m=\left(a_{i_1}, \ldots, a_{i_l}\right): a_t$$
for all $\left{i_1, \ldots, i_l\right} \subset\left{i_1, \ldots, i_n\right}, t, m \in\left{i_1, \ldots, i_n\right} \backslash\left{i_1, \ldots, i_n\right}$.
Often, a minimality condition is imposed on $a_1, \ldots, a_n$ as a set of generators of the ideal it generates in $R$.

Clearly, any permutable regular sequence is a $d$-sequence, but the latter is a lot more flexible to move around as shown in [70, Section 6]. There is an interesting list of examples of ideals having a $d$-sequence as a set of generators ([79]). Here, the main focus is on the following result.

## 数学代写|交换代数代写Commutative Algebra代考|Special properties (survey)

In this section, one surveys a few situations as regards notable properties, normality and Cohen-Macaulayness being foremost. The totality of such results in this respect is too vast and would easily itself fill up a book. Thus, it is to expect that many interesting pieces will not be found here.

Since this is but a survey, no proofs will be given, for which one counts on the reader’s indulgence. Some of these could actually be reproduced here without much toil, but others are quite harder, involving additional technology not developed in the book.

Proposition 7.3.28 ([69, 14.8], $[123$, Theorem 2.1]). Let $(R, \mathfrak{m})$ denote a local ring and let $I \subset \mathfrak{m}$ be an ideal generated by a regular sequence of length $\geq 2$. Then $\operatorname{Proj}(\mathcal{R}(I))$ is smooth if and only if $R / I$ is regular.

The proof is not difficult, but the result itself will not play a role in this book. However, it should be noted that it forces the ambient ring to be regular and the regular sequence to be a regular subsystem of parameters.

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

C. Huneke ([79]) 介绍了以下概念。它以前由 M. Fiorentini ([57]) 以稍微不同的形式定义。

$$\left(a_{i 1}, \ldots, a_{i l}\right): a_t a_m=\left(a_{i 1}, \ldots, a_{i l}\right): a_t$$

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