  We introduce a method for associating a chain complex to a module over a
combinatorial category, such that if the complex is exact then the module has a
rational Hilbert series. We prove homology--vanishing theorems for these
complexes for several combinatorial categories including: the category of
finite sets and injections, the opposite of the category of finite sets and
surjections, and the category of finite dimensional vector spaces over a finite
field and injections.
  Our main applications are to modules over the opposite of the category of
finite sets and surjections, known as $FS^{op}$ modules. We obtain many
constraints on the sequence of symmetric group representations underlying a
finitely generated $FS^{op}$ module. In particular, we describe its character
in terms of functions that we call character exponentials. Our results have new
consequences for the character of the homology of the moduli space of stable
marked curves, and for the equivariant Kazhdan-Luzstig polynomial of the braid
matroid.
