Pointed simplicial complexes (Q1358710)

Let us consider a minimal free resolution of \(M=R/I\), where \(R=k[X_1, \dots, X_n]\), \(\text{char} (k)=p\), \(I=\) monomial ideal of \(R\). The ranks of the free modules of such a resolution might depend on \(p\), but examples are known in which they do not, such as monomial ideals generated by \(R\)-sequences and stable monomial ideals. Using topological tools, the author produces a family of monomial ideals, whose Betti numbers do not depend on \(p\), which includes all the previous examples. In fact to any monomial ideal \(J\) one can attach a square free monomial ideal \(I\), having a corresponding simplicity complex \(\Delta\). If there exists a vertex \(y\) such that: \[ 0\to \widetilde H_i \bigl( \Delta/ \{y\} \bigr)\to \widetilde H_i (\Delta) \to \widetilde H_{i-1} (\text{link } y)\to 0 \] is exact for all \(i\), then the complex is called a ``pointed complex''. This paper investigates how such a property, checked in the associated complex \(\Delta\) and in its sub complexes, is linked to the fact that \(M\)'s Betti numbers do not depend on \(p\).