2013/08/06 by David Cook II, Cook, David
Mathematics · #05C15 #05E45 #06A12 #13D02 #13F55 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.CO #msc:05C15 #msc:05E45 #msc:06A12 #msc:13D02 #msc:13F55
paper · pdf · doi:10.48550/arxiv.1308.1299
34 pages, 8 figures
arxiv created 2013/08/06 · arxiv updated 2013/08/07
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-Söderberg decompositions of both the ideal and its quotient. We also give explicit formulæ 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.