A remark on the local cohomology modules of a union of disjoint matroids
Let \(\Delta\) denote a simplicial complex on the vertex set \(\{1,\ldots,n\}\). Let \(S = K[x_1,\ldots,x_n]\) be the polynomial ring in \(n\) variables over the field \(K\). Let \(S/I\) denote the Stanley-Reisner ring, where \( I = I_{\Delta}\) denotes the Stanley-Reisner ideal of \(\Delta\). In recent research there is much attention about the Cohen-Macaulay-ness (Buchsbaum-ness, \(k\)-Buchsbaum-ness, generalized Cohen-Macaulay-ness etc.) of \(S/I^{(t)}\) for various \(t\geq 1\) in terms of combinatorial data of \(\Delta\). Here \(I^{(t)}, t \geq 1,\) denotes the \(t\)-symbolic power of \(I\). As a main result the author proves a generalization of results of the author's joint work with \textit{Y. Nakamura} [Nagoya Math. J. 213, 127--140 (2014; Zbl 1291.13041)] about the \(k\)-Buchsbaum-ness of \(\Delta\), a union of disjoint matroids. As an application it follows that \(\Delta\) is a matroid if and only if \(S/I^{(t)}\) is \(k\)-Buchsbaum for some \(t \geq \max\{3, k/2 +1\}\) (resp. equivalently for all \(t \geq 1\)).
scientific article; zbMATH DE number 6644454