Buchsbaumness of ordinary powers of two-dimensional square-free monomial ideals
Let \(S=k[x_1,\dots,x_n]\) denote the polynomial ring in \(n\) variables of a field \(k\). Let \(I\) denote the Stanley-Reisner ring of a pure one-dimensional simplicial complex \(\Delta\). The subject of this paper is the investigation when \(S/I^r\) is a Buchsbaum ring. The following results are shown: (1) Let \(n\geq 4\) and \(r\geq 4\), then \(S/I^r\) is a Buchsbaum ring if and only if \(\Delta\) is a square. (2) Let \(n\geq 4\), then \(S/I^3\) is a Buchsbaum ring is and only if \(\Delta\) is a square or a pentagon. (3) With the notation above and \(n\geq 4\) the following conditions are equivalent: (i) \(S/I^2\) is Buchsbaum. (ii) \(\Delta\) is connected, does not contain any triangle of edges, and is claw-free (that is \(\Delta\) has no induced bipartite subgraph with one vertex on one side and three vertices on the other). (iii) \(\Delta\) is a path of length \(n-1\) or a cycle of length \(n\). The main technical tool for the proof is a description of \(\text{Ext}_S^p(S/\mathfrak m_t,S/J)\), where \(\mathfrak m_t=(x_1^t,\dots,x_n^t)S\) and \(J\) is a monomial ideal, in terms of certain simplicial complexes. This is used for proving the Buchsbaum property by the subjectivity criterion. The statements in (1) and (2) are generalizations of corresponding results for Cohen-Macaulay rings shown by \textit{N. C. Minh} and \textit{N. V. Trung} [J. Algebra 322, No. 12, 4219--4227 (2009; Zbl 1206.13028)].
scientific article; zbMATH DE number 5888180
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