The uniform face ideals of a simplicial complex

TL;DR

This paper introduces the uniform face ideal of a simplicial complex based on a vertex colouring, analyzes its algebraic properties, and provides explicit formulas for various invariants, revealing conditions for linear resolutions.

Contribution

It defines the uniform face ideal with respect to vertex colourings, characterizes when it has a linear resolution, and offers explicit formulas for its algebraic invariants.

Findings

The ideal has a linear resolution iff the colouring satisfies a nesting property.

Explicit minimal cellular resolutions are constructed for nested colourings.

Provides formulas for Betti numbers, codimension, dimension, and other invariants.

Abstract

We define the uniform face ideal of a simplicial complex with respect to an ordered proper vertex colouring of the complex. This ideal is a monomial ideal which is generally not squarefree. We show that such a monomial ideal has a linear resolution, as do all of its powers, if and only if the colouring satisfies a certain nesting property. In the case when the colouring is nested, we give a minimal cellular resolution supported on a cubical complex. From this, we give the graded Betti numbers in terms of the face-vector of the underlying simplicial complex. Moreover, we explicitly describe the Boij-S\"oderberg decompositions of both the ideal and its quotient. We also give explicit formul\ae\ for the codimension, Krull dimension, multiplicity, projective dimension, depth, and regularity. Further still, we describe the associated primes, and we show that they are persistent.