Cohen-Macaulay Simplicial Complexes of Degree $k$

Rahim Rahmati-Asghar

Bulletin of the Malaysian Mathematical Society, Tome 37 (2014) no. 1

Résumé

For a positive integer $k$ a class of simplicial complexes, to be denoted by CM($k$), is introduced. This class generalizes Cohen-Macaulay simplicial complexes. In analogy with the Cohen-Macaulay complexes, we give some homological and combinatorial properties of CM($k$) complexes. It is shown that the complex $Δ$ is CM($k$) if and only if $I_{Δ^\vee}$, the Stanley-Reisner ideal of the Alexander dual of $Δ$, has a $k$-resolution, i.e. $\beta_{i.j}(I_{Δ^\vee})=0$ unless $j=ik+q$, where $q$ is the degree of $I_{Δ^\vee}$. As a main result, we characterize all bipartite graphs whose independence complexes are CM($k$) and show that an unmixed bipartite graph is CM($k$) if and only if it is pure $k$-shellable. Our result improves a result due to Herzog and Hibi and also a result due to Villarreal.

Détail

Classification : Primary 13H10; Secondary 05C75