Alexander duality and Stanley depth of multigraded modules

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.