On the \( h \)-triangles of sequentially \((S _r)\) simplicial complexes via algebraic shifting
As the title suggests, the paper under review studies the \(h\)-triangles of sequentially \((S_r)\) simplicial complexes via algebraic shifting. The notion of a sequentially (\( S_r\)) simplicial complex, introduced by \textit{H. Haghighi} et al. [Proc. Am. Math. Soc. 139, No. 6, 1993--2005 (2011; Zbl 1220.13014)], gives a generalization of two properties for simplicial complexes: being sequentially Cohen-Macaulay and satisfying Serre's condition (\( S_r\)). Let \(\Delta\) be a \((d - 1)\)-dimensional \((S_r)\) simplicial complex with \(\Gamma\) (\(\Delta\)) as its algebraic shifting. When \(i\leq r-1\), the \(i\)th pure skeleton \(\Delta^{[i]}\) of \(\Delta\) is Cohen-Macaulay. This fact enables the study of the \(f\)-vectors of the sequential layers of \(\Delta\) and \(\Gamma(\Delta)\), which leads to the main result (Theorem 3.2), namely, algebraic shifting preserves the \(h\)-triangles: \( h_{i,j}(\Delta)= h_{i,j}(\Gamma(\Delta))\) for every \( i \) and \( j \) with \(0 \leq j \leq i \leq r - 1\). Several corollaries, including the study of iterated Betti numbers, are deduced from this main result.
sequentially \((S_r)\) simplicial complex
scientific article; zbMATH DE number 6149023
scientific article