On the asymptotic linearity of Castelnuovo–Mumford regularity

Let R be a standard graded ring over a commutative Noetherian ring with unity. Let I be an arbitrary graded ideal of R and M an arbitrary finitely generated graded R-module. We prove that there exist integers e and ρM(I) such that reg(InM)=ρM(I)n+e for all large n. This generalizes earlier results in the case R is a polynomial ring. Note that standard techniques in the polynomial case cannot be used here.