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- W1913414577 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 $Delta$ 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." @default.
- W1913414577 created "2016-06-24" @default.
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- W1913414577 date "2013-02-24" @default.
- W1913414577 modified "2023-09-27" @default.
- W1913414577 title "On vertex decomposable simplicial complexes and their Alexander duals" @default.
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