On the asymptotic linearity of reduction number

Abstract 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 ( I n M ) , r ( I n M ) and r ( M / I n M ) 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.