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
modules. We obtain many constraints on the sequence of
symmetric group representations underlying a finitely generated
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–Lusztig polynomial of the braid matroid.
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