  Let S be a standard N^r-graded algebra over a local ring A, and let M be a
finitely generated Z^r-graded S-module. We characterize the Cohen-Macaulayness
of M in terms of the vanishing of certain sheaf cohomology modules. As a
consequence, we apply our result to study the Cohen-Macaulayness of multi-Rees
modules (also called Rees modification). Our work extends previous studies on
the Cohen-Macaulayness of multi-Rees algebras.
