  We study the local face modules of triangulations of simplices, i.e., the
modules over face rings whose Hilbert functions are local $h$-vectors. In
particular, we give resolutions of these modules by subcomplexes of Koszul
complexes as well as functorial maps between modules induced by inclusions of
faces. As applications, we prove a new monotonicity result for local
$h$-vectors and new results on the structure of faces in triangulations with
vanishing local $h$-vectors.
