Minimal free resolutions and asymptotic behavior of multigraded regularity

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