On vertex decomposable simplicial complexes and their Alexander duals (Q2795648)

The Alexander dual of a simplicial complex \(\Delta\) is the simplicial complex whose faces are the complements of the nonfaces of \(\Delta\). Alexander duality plays an important role in the study of Stanley-Reisner rings. \textit{J. A. Eagon} and \textit{V. Reiner} [J. Pure Appl. Algebra 130, No. 3, 265--275 (1998; Zbl 0941.13016)] proved that a simplicial complex \(\Delta\) is Cohen-Macaulay if and only if \(I_{\Delta^{\vee}}\) has linear resolution. It is also known that \(\Delta\) is shellable if and only if \(I_{\Delta^{\vee}}\) has linear quotients. Moreover, \textit{J. Herzog} and \textit{T. Hibi} [Nagoya Math. J. 153, 141--153 (1999; Zbl 0930.13018)] proved that \(\Delta\) is sequentially Cohen-Macaulay if and only if \(I_{\Delta^{\vee}}\) is componentwise linear.NEWLINENEWLINEThe aim of the paper under review is to study the Alexander dual of vertex decomposable simplicial complexes. In fact, the authors define the concept of vertex splittable ideals and show that a simplicial complex \(\Delta\) is vertex decomposable if and only if \(I_{\Delta^{\vee}}\) is vertex splittable. 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 the 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.