Some remarks on the Betti numbers of squarefree ideals (Q2704101)

\textbf{Corina Mohorianu (Iaşi)}: The purpose of this paper is to state a new expression for the Betti numbers of a square free lexsegment ideal \(I\subset A\), \(A=K [x_1, \dots, x_n]\) and \(I\) a graded ideal of \(A\). First the authors recall some definitions and results following \textit{A. Aramova}, \textit{J. Herzog} and \textit{T. Hibi} [Math. Z. 228, No. 2, 353-378 (1998; Zbl 0914.13007)], then they express the \(f\)-vector of a lexsegment simplicial complex \(\Delta\) in terms of the \(f\)-vector of \(\Delta_{T_j}\) and the number of a minimal system of generators of \(I_\Delta\), denoted \(\nu(I_\Delta)\), in terms of the \(K\)-dimension of the indicator algebra of \(\Delta\), \(K\{\Delta\}= K[x_1,\dots, x_n]/(I_\Delta, x^2_1, \dots, x^2_n)\).NEWLINENEWLINENEWLINEFinally, the authors study the Betti numbers of a squarefree lexsegment ideal \(I\). For \(\Delta\) the simplicial complex associated to \(I\) and \(\Delta^j\) the simplicial complex corresponding to the squarefree ideal generated by all monomials \(U\) belonging to the unique minimal set of monomial generators of \(I\) such that \(\deg u=j\), they give new formulas for the Betti numbers \(\beta_i(I)\) in terms of the lengths of the simplicial complexes \((\Delta^j)_{T_r}\) for \(r=2,\dots,n\). If \(I_{(r)} =\{u\in G(i):m(u)- \deg(u)+1\leq r\}\) with \(r=0,\dots,n-1\), they determine \(\beta_i(I)\) in terms of the minimal number of generators of \(I_{(r)}\).NEWLINENEWLINENEWLINE\textbf{Alessandro Logar (Trieste)}: Let \(K\) be a field, \(\Delta\) a simplicial complex on the vertex set \(\{1,\dots,n\}\) and \(I_\Delta\) the Stanley-Reisner (monomial and squarefree) ideal associated to \(\Delta\) over \(K\). If \(I\) is a monomial, squarefree ideal of \(K[x_1,\dots,x_n]\), \(I\) is called squarefree lexsegment if the following holds: if \(u \in I\) and \(v\) is a squarefree monomial of the same degree of \(u\) and \(u \leq_{\text{lex}}v\), then \(v \in I\) (where `\(\leq_{\text{lex}}\)'' denotes the lexicographics order). Analogously, a simplicial complex is called a lexsegment simplicial complex if \(I_\Delta\) is a lexsegment ideal.NEWLINENEWLINE The main result of the paper is a formula for the computation of the minimal numbers of generators of \(I_\Delta\) and for the Betti's numbers of \(I_\Delta\) when \(\Delta\) is a lexsegment simplicial complex.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00040].