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.Comment: 18 pages. We have removed Lemma 2.3 of the previous version, since
the proof contained a gap. This deletion does not affect the main results,
while we have revised argument a little (especially in Sections in 2 and 3