On the asymptotic linearity of reduction number

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 $n\gg 0$.