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

TL;DR

This paper investigates the algebraic structure of local face modules in simplicial triangulations, providing resolutions, functorial maps, and new insights into local $h$-vector behavior and face structures.

Contribution

It introduces resolutions of local face modules via Koszul complexes and establishes functorial maps, advancing understanding of local $h$-vectors and face structures.

Findings

Proved a new monotonicity property for local $h$-vectors.

Derived structural results for triangulations with vanishing local $h$-vectors.

Developed resolutions of face modules using subcomplexes of Koszul complexes.

Abstract

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.