  We study equivariant resolutions and local cohomologies of toric sheaves for
affine toric varieties, where our focus is on the construction of new examples
of decomposable maximal Cohen-Macaulay modules of higher rank. A result of
Klyachko states that the category of reflexive toric sheaves is equivalent to
the category of vector spaces together with a certain family of filtrations.
Within this setting, we develop machinery which facilitates the construction of
minimal free resolutions for the smooth case as well as resolutions which are
acyclic with respect to local cohomology functors for the general case. We give
two main applications. First, over the polynomial ring, we determine in
explicit combinatorial terms the Z^n-graded Betti numbers and local cohomology
of reflexive modules whose associated filtrations form a hyperplane
arrangement. Second, for the non-smooth, simplicial case in dimension d >= 3,
we construct new examples of indecomposable maximal Cohen-Macaulay modules of
rank d - 1.
