Hierarchies of Simplicial Complexes via the Bgg-correspondence
نویسنده
چکیده
Via the BGG-correspondence a simplicial complex ∆ on [n] is transformed into a complex of coherent sheaves L̃(∆) on the projective space P. In general we compute the support of each of its cohomology sheaves. When the Alexander dual ∆ is Cohen-Macaulay there is only one such non-zero cohomology sheaf. By considering when this sheaf can be an a’th syzygy sheaf in a locally free resolution, we get a hierarchy of simplicial complexes CM ⊇ · · · ⊇ CMa ⊇ · · · with nice compact criteria for membership. These classes behave nicely with respect to taking links and restrictions. By putting further conditions on the sheaves we get in CMa nice families of simplicial complexes whose f -vector depends only on n, d, and c. When a = 0 these are the bi-Cohen-Macaulay simplicial complexes and when a = c we get Alexander duals of the Steiner systems S(c, d, n). We also show that ∆ is Gorenstein* iff the associated coherent sheaf of ∆ is an ideal sheaf.
منابع مشابه
New methods for constructing shellable simplicial complexes
A clutter $mathcal{C}$ with vertex set $[n]$ is an antichain of subsets of $[n]$, called circuits, covering all vertices. The clutter is $d$-uniform if all of its circuits have the same cardinality $d$. If $mathbb{K}$ is a field, then there is a one-to-one correspondence between clutters on $V$ and square-free monomial ideals in $mathbb{K}[x_1,ldots,x_n]$ as follows: To each clutter $mathcal{C}...
متن کاملVertex Decomposable Simplicial Complexes Associated to Path Graphs
Introduction Vertex decomposability of a simplicial complex is a combinatorial topological concept which is related to the algebraic properties of the Stanley-Reisner ring of the simplicial complex. This notion was first defined by Provan and Billera in 1980 for k-decomposable pure complexes which is known as vertex decomposable when . Later Bjorner and Wachs extended this concept to non-pure ...
متن کاملCohen-Macaulay-ness in codimension for simplicial complexes and expansion functor
In this paper we show that expansion of a Buchsbaum simplicial complex is $CM_t$, for an optimal integer $tgeq 1$. Also, by imposing extra assumptions on a $CM_t$ simplicial complex, we provethat it can be obtained from a Buchsbaum complex.
متن کاملOn a special class of Stanley-Reisner ideals
For an $n$-gon with vertices at points $1,2,cdots,n$, the Betti numbers of its suspension, the simplicial complex that involves two more vertices $n+1$ and $n+2$, is known. In this paper, with a constructive and simple proof, wegeneralize this result to find the minimal free resolution and Betti numbers of the $S$-module $S/I$ where $S=K[x_{1},cdots, x_{n}]$ and $I$ is the associated ideal to ...
متن کامل