Let S be a standard graded polynomial ring over a field K in a finite set of variables, and let m be the graded maximal ideal of S. It is known that for a finitely generated graded S-module M and all integers k 0, the module mkM is componentwise linear. For large k we describe the pattern of the Betti table of mkM when depthM >0. Moreover, we show that for any k 0, mkI has linear quotients if I is a monomial ideal.
