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

Abstract

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 $\mathbf{rt}(I;M)$, is the maximal degree in a minimal generating set of relations of the Rees module $\mathbf{R}(I;M)={\oplus }_{n\ge 0}{I}^{n}M$. It is a well-known invariant that gives a first measure of the complexity of $\mathbf{R}(I;M)$. To help to measure this complexity, we introduce the sifted type of $\mathbf{R}(I;M)$, denoted by $\mathbf{st}(I;M)$, a new invariant which counts the non-zero degrees appearing in a minimal generating set of relations of $\mathbf{R}(I;M)$. Just as the relation type $\mathbf{rt}(I;M/N)$ is closely related to the strong Artin-Rees number $\mathbf{s}(N,M;I)$, it turns out that the sifted type $\mathbf{st}(I;M/N)$ is closely related to the medium Artin-Rees number $\mathbf{m}(N,M;I)$, 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 $\mathbf{m}(N,M;I)$ and $\mathbf{st}(I;M)$ with some examples.