  A foundational principle in the study of modules over standard graded
polynomial rings is that geometric positivity conditions imply vanishing of
Betti numbers. The main goal of this paper is to determine the extent to which
this principle extends to the nonstandard graded case. In this setting, the
classical arguments break down, and the results become much more nuanced. We
introduce a new notion of Castelnuovo-Mumford regularity and employ exterior
algebra techniques to control the shapes of nonstandard graded minimal free
resolutions. Our main result reveals a unique feature in the nonstandard graded
case: the possible degrees of the syzygies of a graded module in this setting
are controlled not only by its regularity, but also by its depth. As an
application of our main result, we show that, given a simplicial projective
toric variety and a module M over its coordinate ring, the multigraded Betti
numbers of M are contained in a particular polytope when M satisfies an
appropriate positivity condition.
