On the asymptotic linearity of reduction number

Let R be a standard graded algebra over an infinite field K and M a finitely generated Z-graded R-module. For any graded ideal I⊆R+ of R, we show that the functions D(InM),r(InM) and r(M/InM) are all asymptotically linear. Here r(
- ) and D(
- ) stand for the reduction number and the maximal degree of minimal generators of a graded module
- respectively.