  We apply Miller's theory on multigraded modules over a polynomial ring to the
study of the Stanley depth of these modules. Several tools for Stanley's
conjecture are developed, and a few partial answers are given. For example, we
show that taking the Alexander duality twice (but with different "centers") is
useful for this subject. Generalizing a result of Apel, we prove that Stanley's
conjecture holds for the quotient by a cogeneric monomial ideal.
