On reduction number of products of ideals

TL;DR

This paper investigates the asymptotic behavior of the reduction number of modules formed by products of graded ideals in a standard graded algebra over an infinite field, providing new insights into their growth patterns.

Contribution

It introduces the asymptotic analysis of the reduction number for modules involving products of multiple graded ideals, a novel perspective in the study of graded algebra modules.

Findings

Asymptotic formulas for the reduction number functions are established.

The behavior of reduction numbers stabilizes under certain conditions.

New bounds and growth patterns for reduction numbers are derived.

Abstract

Let $R$ be a standard graded algebra over an infinite field $\mathbb{K}$ and $M$ a finitely generated $\ZZ$-graded $R$-module. Let $I_1,\ldots I_m$ be graded ideals of $R$. The functions $r(M/I_1^{a_1}\ldots I_m^{a_m}M)$ and $r(I_1^{a_1}\ldots I_m^{a_m}M)$ are investigated and their asymptotical behaviours are given. Here $r(\bullet)$ stands for the reduction number of a finitely graded module $\bullet$.