标题： 滤过次数的方程的Rees模及其与Artin-Rees数的联系 显示英文标题

标题： Sifted degrees of the equations of the Rees module and their connection with the Artin-Rees numbers

摘要： 设$A$为一个诺特环，$I$为$A$的一个理想，$N\subset M$为有限生成的$A$-模。 $I$与$M$的关系类型，记为${\bf rt}\,(I;M)$，是 Rees 模块${\bf R}(I;M)=\oplus_{n\geq 0}I^nM$的关系的最小生成集中的最大次数。这是一个已知的不变量，用于衡量${\bf R}(I;M)$的复杂性的初步度量。 为了帮助衡量这种复杂性，我们引入了类型${\bf R}(I;M)$的筛选形式，记为${\bf st}\,(I;M)$，这是一个新的不变量，它统计了${\bf R}(I;M)$的关系的最小生成集中的非零次数。 正如关系类型${\bf rt}\,(I;M/N)$与强Artin-Rees数${\bf s}\,(N,M;I)$密切相关一样，结果表明筛选类型${\bf st}\,(I;M/N)$与中等Artin-Rees数${\bf m}\,(N,M;I)$密切相关，这是一个新的不变量，位于$(N,M;I)$的弱Artin-Rees数和强Artin-Rees数之间。我们通过一些例子来说明${\bf m}\,(N,M;I)$和${\bf st}\,(I;M)$的含义、兴趣以及相互联系。 显示英文摘要

摘要： Let $A$ be a noetherian ring, $I$ an ideal of $A$ and $N\subset M$ finitely generated $A$-modules. The relation type of $I$ with respect to $M$, denoted by ${\bf rt}\,(I;M)$, is the maximal degree in a minimal generating set of relations of the Rees module ${\bf R}(I;M)=\oplus_{n\geq 0}I^nM$. It is a well-known invariant that gives a first measure of the complexity of ${\bf R}(I;M)$. To help to measure this complexity, we introduce the sifted type of ${\bf R}(I;M)$, denoted by ${\bf st}\,(I;M)$, a new invariant which counts the non-zero degrees appearing in a minimal generating set of relations of ${\bf R}(I;M)$. Just as the relation type ${\bf rt}\,(I;M/N)$ is closely related to the strong Artin-Rees number ${\bf s}\,(N,M;I)$, it turns out that the sifted type ${\bf st}\,(I;M/N)$ is closely related to the medium Artin-Rees number ${\bf m}\,(N,M;I)$, a new invariant which lies in between the weak and the strong Artin-Rees numbers of $(N,M;I)$. We illustrate the meaning, interest and mutual connection of ${\bf m}\,(N,M;I)$ and ${\bf st}\,(I;M)$ with some examples.