A colorful Hochster formula and universal parameters for face rings

TL;DR

This paper extends Hochster's formula to a colorful setting for vertex-colored simplicial complexes and explores universal parameters for face rings that reveal algebraic properties and symmetry invariance.

Contribution

It introduces a colorful Hochster formula applicable to vertex-colored complexes and studies universal parameters that encode depth and symmetry properties of face rings.

Findings

Generalized Hochster's formula to colorful complexes

Identified universal parameters with symmetry-invariant properties

Conjectured shape of resolutions over these parameters

Abstract

This paper has two related parts. The first generalizes Hochster's formula on resolutions of Stanley-Reisner rings to a colorful version, applicable to any proper vertex-coloring of a simplicial complex. The second part examines a universal system of parameters for Stanley-Reisner rings of simplicial complexes, and more generally, face rings of simplicial posets. These parameters have good properties, including being fixed under symmetries, and detecting depth of the face ring. Moreover, when resolving the face ring over these parameters, the shape is predicted, conjecturally, by the colorful Hochster formula.