Generalizing the Borel property

We introduce the notion of Q-Borel ideals: ideals which are closed under the Borel moves arising from a poset Q. We study decompositions and homological properties of these ideals, and offer evidence that they interpolate between Borel ideals and arbitrary monomial ideals.Comment: 19 pages, 1 figur

The uniform face ideals of a simplicial complex

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.Comment: 34 pages, 8 figure

Poset shifted matroids and graphs

We generalize the notion of shiftedness in simplicial complexes, matroids, and graphs. Using this generalization, called $P$-shiftedness, we give a condition for a matroid to be transversal in terms of a poset $P$. Our result both generalizes and recovers a similar result that characterizes all shifted matroids as transversal. Moreover, we characterize certain shifted and $P$-shifted simple graphic matroids. We also explore the properties of $P$-shifted graphs and classify certain $P$-shifted graph families