  In this paper, we study ideals $I$ whose linear strand can be supported on a
regular CW complex. We provide a sufficient condition for the linear strand of
an arbitrary subideal of $I$ to remain supported on an easily described
subcomplex. In particular, we prove that a certain class of rainbow monomial
ideals always have linear strand supported on a regular CW complex, including
any initial ideal of the ideal of maximal minors of a generic matrix. We also
provide a sufficient condition for these ideals to have linear resolution,
which is also an equivalence under mild assumptions. We then employ a result of
Almousa, Fl\o ystad, and Lohne to apply these results to polarizations of
Artinian monomial ideals. We conclude with further questions relating to
cellularity of certain classes of squarefree monomial ideals and the
relationship between initial ideals of maximal minors and algebra structures on
certain resolutions.
