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.

In this paper, we explain the regularity, projective dimension and depth of edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a C 5 -free vertex decomposable graph G, reg(R/I(G)) = c G , where c G is the maximum number of 3-disjoint edges in G. Moreover for this class of graphs we characterize pd(R/I(G)) and depth(R/I(G)). As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.

Communicated by A. J. SommeseFor a simplicial complex ∆, we introduce a simplicial complex attached to ∆, called the expansion of ∆, which is a natural generalization of the notion of expansion in graph theory. We are interested in knowing how the properties of a simplicial complex and its Stanley-Reisner ring relate to those of its expansions. It is shown that taking expansion preserves vertex decomposable and shellable properties and in some cases Cohen-Macaulayness. Also it is proved that some homological invariants of Stanley-Reisner ring of a simplicial complex relate to those invariants in the Stanley-Reisner ring of its expansions.

In this paper, we compare Krull dimension, Gorenstein injective dimension and injective dimension of a module in several cases. In fact, we establish some generalizations of the Bass formula. To this end, we generalize the Grothendieck non-vanishing theorem to a class of modules larger than finitely generated modules. Mathematics Subject Classification (2000). 13D45, 13C15, 13E05.

No abstract