On Vertex Decomposable Simplicial Complexes and Their Alexander Duals

Abstract

Abstract. 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 ∆ is vertex decomposable if and only if I ∆ ∨ 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 ∆ , when ∆ 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.

Betti splitting via componentwise linear ideals

“…As a consequence of Theorem 3.3, we recover a result due to Moradi and Kosh-Ahang [17], proving that the Alexander dual of a vertex-decomposable simplicial complex admits a particular kind of splitting, the so-called x i -splitting (Corollary 4.3). In Corollary 4.5 we prove a Betti splitting result for shellable and constructible simplicial complexes, showing that in general they do not admit x i -splitting (Example 4.4).…”

“…Many authors wrote papers applying the Eliahou-Kervaire technique to the resolution of special classes of monomial ideals (see e.g. [4], [5], [6], [8], [17], [19], [23], [24], [25]). Francisco, Hà and Van Tuyl proved in [7,Corollary 2.4] that if J and K have a linear resolution, then I = J + K is a Betti splitting of I.…”

Recursive Betti numbers for Cohen–Macaulay d -partite clutters arising from posets

“…Let i be the vertex in the definition of weakly vertex decomposable simplicial complex. Since del ∆ (i) is Cohen-Macaulay, then it is pure and i is a shedding vertex (see for instance [25] In the following example we show that the converse of the previous proposition does not hold, even in the Cohen-Macaulay case. Figure 3, due to Hachimori [18].…”

Vertex decomposability and weakly polymatroidal ideals

“…Moradi and Khosh-Ahang [20] defined the notion of vertex splittable ideal which is an algebraic analog of the vertex decomposability property of a simplicial complex and was defined as follows: Definition 1.1. A monomial ideal I in R is called vertex splittable if it can be obtained by the following recursive procedure:…”

“…Herzog, Hibi and Zheng [13] proved that the simplicial complex ∆ is shellable if and only if I ∨ has linear quotients. Recently Moradi and Khosh-Ahang [20] proved that the simplicial complex ∆ is vertex decomposable if and only if I ∨ is vertex splittable and also they showed that every vertex splittable ideal has linear quotients. Hence we have the following implications:…”