386 research outputs found
Cohen-Macaulay graphs and face vectors of flag complexes
We introduce a construction on a flag complex that, by means of modifying the
associated graph, generates a new flag complex whose -factor is the face
vector of the original complex. This construction yields a vertex-decomposable,
hence Cohen-Macaulay, complex. From this we get a (non-numerical)
characterisation of the face vectors of flag complexes and deduce also that the
face vector of a flag complex is the -vector of some vertex-decomposable
flag complex. We conjecture that the converse of the latter is true and prove
this, by means of an explicit construction, for -vectors of Cohen-Macaulay
flag complexes arising from bipartite graphs. We also give several new
characterisations of bipartite graphs with Cohen-Macaulay or Buchsbaum
independence complexes.Comment: 14 pages, 3 figures; major updat
Locally standard torus actions and h'-vectors of simplicial posets
We consider the orbit type filtration on a manifold with locally standard
action of a compact torus and the corresponding homological spectral sequence
. If all proper faces of the orbit space are acyclic,
and the free part of the action is trivial, this spectral sequence can be
described in full. The ranks of diagonal terms are equal to the -numbers of
the Buchsbaum simplicial poset dual to . Betti numbers of depend
only on the orbit space but not on the characteristic function. If is a
slightly different object, namely the model space where
is a cone over Buchsbaum simplicial poset , we prove that . This gives a topological evidence for the
fact that -numbers of Buchsbaum simplicial posets are nonnegative.Comment: 21 pages, 3 figures + 1 inline figur
Stanley-Reisner rings of Buchsbaum complexes with a free action by an abelian group
We consider simplicial complexes admitting a free action by an abelian group.
Specifically, we establish a refinement of the classic result of Hochster
describing the local cohomology modules of the associated Stanley--Reisner
ring, demonstrating that the topological structure of the free action extends
to the algebraic setting. If the complex in question is also Buchsbaum, this
new description allows for a specialization of Schenzel's calculation of the
Hilbert series of some of the ring's Artinian reductions. In further
application, we generalize to the Buchsbaum case the results of Stanley and
Adin that provide a lower bound on the -vector of a Cohen-Macaulay complex
admitting a free action by a cyclic group of prime order
- …