On the asymptotic linearity of reduction number

TL;DR

This paper proves that the reduction number and maximal degree of minimal generators of powers of a graded module grow linearly with the power, introducing a generalized regularity function that also exhibits linearity asymptotically.

Contribution

It establishes the asymptotic linearity of reduction numbers and introduces a generalized regularity function for graded modules over Noetherian rings.

Findings

Reduction number of $I^nM$ grows linearly with $n$ for large $n$.

Maximal degree of minimal generators of $I^nM$ grows linearly with $n$ for large $n$.

Generalized regularity function $ ext{Gamma}(I^nM)$ is linear in $n$ asymptotically.

Abstract

Let $R$ be a standard graded Noetherian algebra over an infinite field $K$ and $M$ a finitely generated $\mathbb{Z}$-graded $R$-module. Then for any graded ideal $I\subseteq R_+$ of $R$, we show that there exist integers $e_1\geq e_2$ such that $r(I^nM)=\rho_I(M)n+e_1$ and $D(I^nM)=\rho_I(M)n+e_2$ for $n\gg0$. Here $r(M)$ and $D(M)$ denote the reduction number of $M$ and the maximal degree of minimal generators of $M$ respectively, and $\rho_I(M)$ is an integer determined by both $M$ and $I$. We introduce the notion of generalized regularity function $\Gamma$ for a standard graded algebra over a Noetherian ring and prove that $\Gamma(I^nM)$ is also a linear function in $n$ for $n\gg 0$.