  For a finitely generated graded module $M$ over a positively-graded
commutative Noetherian ring $R$, the second author established in 1999 some
restrictions, which can be formulated in terms of the Castelnuovo regularity of
$M$ or the so-called $a^*$-invariant of $M$, on the supporting degrees of a
graded-indecomposable graded-injective direct summand, with associated prime
ideal containing the irrelevant ideal of $R$, of any term in the minimal
graded-injective resolution of $M$. Earlier, in 1995, T. Marley had established
connections between finitely graded local cohomology modules of $M$ and local
behaviour of $M$ across $\Proj(R)$. The purpose of this paper is to present
some multi-graded analogues of the above-mentioned work.
