Sifted degrees of the equations of the Rees module and their connection with the Artin-Rees numbers
Philippe Gimenez, Francesc Planas-Vilanova
TL;DR
The paper introduces the sifted type ${\bf st}(I;P)$ and the medium Artin–Rees number ${\bf m}(N,M;I)$, situating them alongside the classical relation type ${\bf rt}(I;P)$ and the strong/weak Artin–Rees numbers ${\bf s}(N,M;I)$ and ${\bf w}(N,M;I)$. It establishes a chain of inequalities ${\bf w}\le{\bf m}\le{\bf s}\le{\bf st}(I;M/N)\le{\bf rt}(I;M/N)$ and relates ${\bf st}$ to ${\bf m}$ via exact sequences of effective relations $E(I;P)_n$, thereby connecting degrees of defining equations of Rees modules to Artin–Rees data. A central contribution is the Fibonacci-like construction showing that the set of sifted degrees ${\rm SD}(I)$ can realize rich patterns, with ${\bf st}(I;M)$ potentially far smaller than ${\bf rt}(I;M)$; this is illustrated through explicit monomial and two-variable ideals. The paper closes with classical and new examples that demonstrate the range of possible invariant values and sharpen our understanding of how Rees-module equations reflect Artin–Rees behavior in diverse algebraic contexts.
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 ${\bf rt}\,(I;M)$, 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.