Resolutions of local face modules, functoriality, and vanishing of local h-vectors

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.