2,915 research outputs found
Regularity of squarefree monomial ideals
We survey a number of recent studies of the Castelnuovo-Mumford regularity of
squarefree monomial ideals. Our focus is on bounds and exact values for the
regularity in terms of combinatorial data from associated simplicial complexes
and/or hypergraphs.Comment: 23 pages; survey paper; minor changes in V.
Independence complexes and incidence graphs
We show that the independence complex of the incidence graph of a hypergraph is homotopy equivalent to the combinatorial Alexander dual of the independence complex of the hypergraph, generalizing a result of Csorba. As an application, we refine and generalize a result of Kawamura on a relation between the homotopy types of the independence complex and the edge covering complex of a graph
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
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