Journal of Commutative Algebra

The uniform face ideals of a simplicial complex

David Cook II

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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.

Article information

J. Commut. Algebra, Volume 11, Number 2 (2019), 175-224.

First available in Project Euclid: 24 June 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13F55: Stanley-Reisner face rings; simplicial complexes [See also 55U10]
Secondary: 05E45: Combinatorial aspects of simplicial complexes 13D02: Syzygies, resolutions, complexes 05C15: Coloring of graphs and hypergraphs 06A12: Semilattices [See also 20M10; for topological semilattices see 22A26]

Monomial ideal linear resolution cellular resolution Betti numbers simplicial complex vertex colouring face ideal


II, David Cook. The uniform face ideals of a simplicial complex. J. Commut. Algebra 11 (2019), no. 2, 175--224. doi:10.1216/JCA-2019-11-2-175.

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