Supporting degrees of multi-graded local cohomology modules

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.