On the asymptotic linearity of Castelnuovo–Mumford regularity

Abstract

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 ρ M ( I ) such that reg ( I n M ) = ρ M ( 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.