  In recent years, two different multigraded variants of Castelnuovo-Mumford
regularity have been developed, namely multigraded regularity, defined by the
vanishing of multigraded pieces of local cohomology modules, and the resolution
regularity vector, defined by the multidegrees in a minimal free resolution. In
this paper, we study the relationship between multigraded regularity and the
resolution regularity vector. Our method is to investigate multigraded variants
of the usual a*-invariant. This, in particular, provides an effective approach
to examining the vanishing of multigraded pieces of local cohomology modules
with respect to different graded irrelevant ideals.
