Sequentially Cohen-Macaulay path ideals of cycles (Q2919574)

In the paper there are 2 main theorems and a counter example to an earlier conjecture from \textit{W. Bruns} and \textit{T. Hibi} [Commun. Algebra 23, No. 4, 1201--1217 (1995; Zbl 0823.13015)].NEWLINENEWLINE Take an integer \(n\geq t\geq 2\) and let \(G\) be a directed graph. A sequence \(x_{i_1},\dots,x_{i_t}\) of distinct vertices, is called a path of length \(t\) if there are \(t-1\) distinct directed edges \(e_1,\dots,e_{t-1}\) where \(e_j\) is a directed edge from \(x_{i_j}\) to \(x_{i_{j+1}}\). Then the path ideal of \(G\) of length \(t\) is the monomial ideal \(I_t(G)=(x_{i_1}\cdots x_{i_t}:x_{i_1},\dots,x_{i_t}\) is a path of length \(t\) in \(G\)) in the polynomial ring \(R=k[x_1,\dots,x_n]\) over a field \(k\). \(C_n\) denotes the \(n\)-cycle.NEWLINENEWLINE Theorem. Let \(t\geq 3\). Then \(I_t(C_n)\) is unmixed if and only if \(t\leq n\leq\lfloor 3t/2\rfloor +1\) or \(n=2t+1\).NEWLINENEWLINE Theorem. Let \(t\geq 3\). Then \(R/I_t(C_n)\) is sequentially Cohen-Macaulay if and only if \(n=t\) or \(t+1\) or \(2t+1\).NEWLINENEWLINE The conjecture was that there exists no 2-dimensional simplicial complex \(\Delta\) on the vertex set \(V\) with \(| V|\geq 9\) and 2 does not divide \(| V|\) such that \(k[\Delta]\) has a 3-pure, but not 3-linear resolution. Now the authors give an example which shows that this conjecture is not true.