The Buchsbaum property of symbolic powers of Stanley-Reisner ideals of dimension 1
Let \(\Delta\) be a pure \(1\)-dimensional simplicial complex on a vertex set \([n]=\{1,2,\dots,n\}\). Let \(S=k[x_1,\dots,x_n]\) be a standard graded polynomial ring over an infinite field. We consider the Stanley-Reisner ring \(S/I\), where \(I=I_\Delta\) is the Stanley-Reisner ideal. It is known that \(S/I\) is always Buchsbaum and it is Cohen-Macaulay if and only if \(\Delta\) is connected. In this paper, the authors consider the problem to characterize Buchsbaumness of \(S/I^{(r)}\) for \(r\in\mathbb N\), where \(I^{(r)}\) denotes the \(r\)th symbolic power of \(I\). The main result (Theorem~3.7) states that (1) \(S/I^{(2)}\) is Buchsbaum if and only if \(\Delta\) is connected, (2) \(S/I^{(3)}\) is Buchsbaum if and only if \(\text{diam}(\Delta)\leq 2\), and (3) for \(r>3\) if \(S/I^{(r)}\) is Buchsbaum, then it is Cohen-Macaulay. Here we mean by \(\text{diam}(\Delta)\) the diameter of \(\Delta\), which means the maximal value of the distance between any two vertices in \(\Delta\). On the other hand, \(S/I^{(r)}\) is of dimension~2 and generalized Cohen-Macaulay, i.e., the local cohomology module \(H^1_{\mathfrak m}(S/I^{(r)})\) is of finite length, where \(\mathfrak m=(x_1,\dots,x_m)S\) is the homogeneous maximal ideal. Thus there exists \(k\in\mathbb N\) such that \(\mathfrak m^kH^1_{\mathfrak m}(S/I^{(r)})=0\). The authors computes the smallest such number \(k\) (Theorem~3.8).
scientific article; zbMATH DE number 5807457