  Let S be a standard N^k-graded polynomial ring over a field. Let I be a
multigraded homogeneous ideal in S and let M be a finitely generated Z^k-graded
S-module. We prove that the resolution regularity, a multigraded variant of
Castelnuovo-Mumford regularity, of I^nM is asymptotically a linear function.
This shows that the well known Z-graded phenomenon carries to multigraded
situation.
