  We study global primary decompositions in the category of sheaves on a scheme
which are equivariant under the action of an algebraic group. We show that
equivariant primary decompositions exist if the group is connected. As main
application we consider the case of varieties which are quotients of a
quasi-affine variety by the action of a diagonalizable group and thus admit a
homogeneous coordinate ring, such as toric varieties. Comparing these
decompositions with primary decompositions of graded modules over the
homogeneous coordinate ring, we show that these are equivalent if the action of
the diagonalizable group is free. We give some specific examples for the case
of toric varieties.
