TY - JOUR
AU - Moradi, Somayeh
AU - Khosh-Ahang, Fahimeh
PY - 2016/03/07
Y2 - 2024/06/14
TI - On Vertex Decomposable Simplicial Complexes and Their Alexander Duals
JF - MATHEMATICA SCANDINAVICA
JA - Math. Scand.
VL - 118
IS - 1
SE - Articles
DO - 10.7146/math.scand.a-23295
UR - https://www.mscand.dk/article/view/23295
SP - 43-56
AB - In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex $\Delta$ is vertex decomposable if and only if $I_{\Delta^{\vee}}$ is a vertex splittable ideal. Moreover, the properties of vertex splittable ideals are studied. As the main result, it is proved that any vertex splittable ideal has a Betti splitting and the graded Betti numbers of such ideals are explained with a recursive formula. As a corollary, recursive formulas for the regularity and projective dimension of $R/I_{\Delta}$, when $\Delta$ is a vertex decomposable simplicial complex, are given. Moreover, for a vertex decomposable graph $G$, a recursive formula for the graded Betti numbers of its vertex cover ideal is presented. In special cases, this formula is explained, when $G$ is chordal or a sequentially Cohen-Macaulay bipartite graph. Finally, among the other things, it is shown that an edge ideal of a graph is vertex splittable if and only if it has linear resolution.
ER -