A note on the \(k\)-Buchsbaum property of symbolic powers of Stanley-Reisner ideals
Let \(S=F[x_1,\dots,x_n]\) be a polynomial ring over a field \(F.\) Let \(S/I_{\Delta}\) be the Stanley-Reisner ring associated to a pure simplicial complex \(\Delta,\) of dimension \(1,\) on the vertex set \(\{1,\dots,n\}.\) The goal of the paper is to study the \(k\)-Buchsbaum property of \(S/I^{(r)},\) where \(I^{(r)}\) is the \(r\)-th symbolic power of \(I=I_{\Delta}.\) The condition for \(S/I^{(r)}\) to be \(k\)-Buchsbaum is equivalent to saying that \(k\) is the minimal number satisfying \(m^kH^1_m(S/I^{(r)})=0,\) where \(m=(x_1,\dots,x_n)+I.\) In the main theorem the function \[ k(r):=\min\{k\mid m^kH^1_m(S/I^{(r)}=0\} \] is computed when \(\Delta\) is not connected. The connected case was solved in a previous paper of the same authors.
scientific article; zbMATH DE number 5950742