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

Matt Larson, Sam Payne, Alan Stapledon

TL;DR

The paper introduces local face modules $L(\Gamma,E)$ for quasi-geometric homology triangulations $\Gamma$ of simplices, whose local $h$-vectors are their Hilbert functions. It provides an explicit Koszul-type resolution of $L(\Gamma,E)$ and constructs natural functorial maps $L(\Gamma,E)\to L(\Gamma,E')$ for face inclusions, enabling a functorial framework for these invariants. A key result is the monotonicity of local $h$-vectors when $\sigma(E)=\sigma(E')$, and the authors develop structural results on triangulations with vanishing local $h$-vectors via restrictions and the notion of $U$-pyramids. The methods apply to quasi-geometric triangulations (not only geometric ones) and yield new insights into the monodromy conjectures, with concrete illustrations like the triforce example demonstrating the resolution and maps.

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.

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

TL;DR

The paper introduces local face modules for quasi-geometric homology triangulations of simplices, whose local -vectors are their Hilbert functions. It provides an explicit Koszul-type resolution of and constructs natural functorial maps for face inclusions, enabling a functorial framework for these invariants. A key result is the monotonicity of local -vectors when , and the authors develop structural results on triangulations with vanishing local -vectors via restrictions and the notion of -pyramids. The methods apply to quasi-geometric triangulations (not only geometric ones) and yield new insights into the monodromy conjectures, with concrete illustrations like the triforce example demonstrating the resolution and maps.

Abstract

We study the local face modules of triangulations of simplices, i.e., the modules over face rings whose Hilbert functions are local -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 -vectors and new results on the structure of faces in triangulations with vanishing local -vectors.
Paper Structure (7 sections, 14 theorems, 51 equations, 1 figure)

This paper contains 7 sections, 14 theorems, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Triangulations of simplices
  4. Face rings and special l.s.o.p.s
  5. A resolution of the local face module
  6. Functorial properties of local face modules
  7. Restrictions of local face modules

Key Result

Theorem 1.2

There is an exact sequence of graded $k[\mathop{\mathrm{lk}}\nolimits_{\Gamma}(E)]$-modules where, for $S= \{v_{i_0}, \ldots, v_{i_k}\}$, with $i_0 < \cdots < i_k$, the differential restricted to $I_S$ is $\oplus_{j = 0}^k (-1)^j \lambda_{i_j}$.

Figures (1)

  • Figure 1: The double complex obtained by taking the Koszul resolution of \ref{['eq:quotientedcomplex']}.

Theorems & Definitions (37)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 27 more