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2019 The uniform face ideals of a simplicial complex
David Cook II
J. Commut. Algebra 11(2): 175-224 (2019). DOI: 10.1216/JCA-2019-11-2-175

Abstract

We introduce 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 though it is generated in a single degree. 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-Soderberg decompositions of both the ideal and its quotient. We also give explicit formulae 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.

Citation

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David Cook II. "The uniform face ideals of a simplicial complex." J. Commut. Algebra 11 (2) 175 - 224, 2019. https://doi.org/10.1216/JCA-2019-11-2-175

Information

Published: 2019
First available in Project Euclid: 24 June 2019

zbMATH: 07080074
MathSciNet: MR3973136
Digital Object Identifier: 10.1216/JCA-2019-11-2-175

Subjects:
Primary: 13F55
Secondary: 05C15, 05E45, 06A12, 13D02

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.11 • No. 2 • 2019
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