Generalized complete intersections with linear resolutions (Q948896)

Let \(S = K[X_1,\ldots ,X_n]\) be a standard graded polynomial ring over a field \(K\) and \(\mathfrak {m} = (X_1,\ldots ,X_n)\). The authors give a combinatorial characterization of the simplicial complexes whose Stanley-Reisner ideals are generalized complete intersections and all powers of the ideals have linear resolutions. The main result of the paper under review shows for a \(d\)-dimensional pure simplicial complex \(\Delta\) with core\((\Delta) = \Delta\) the following conditions are equivalent: (1) The Stanley-Reisner ring \(K[\Delta] = S/I_{\Delta}\) is a generalized complete intersection and \(I_{\Delta}^{\ell}\) has a linear resolution for all \(\ell \geq 1\); (2) \(\Delta\) is a finite set of points, or otherwise \(\Delta\) is a disjoint union of paths \(\Gamma_1, \ldots , \Gamma_s\) of arbitrary length if \(d = 1\), or is a disjoint union of the following types of subcomplexes if \(d \geq 2\), type 1. \(\langle F,G \rangle\), where \(F\) and \(G\) are facets such that \(|F \setminus G| = |G \setminus F| = 1\), type 2. \(\langle H \rangle\), where \(H\) is a facet; (3) The power of the Stanley-Reisner ideal \(I_{\Delta}^{\ell}\) has finite local cohomology and a linear resolution for all \(\ell \geq 1\). If one of the above conditions holds then we have the following isomorphism as \(K\)-vector spaces \[ [H_{\mathfrak m}^{i}(S/I_{\Delta}^{\ell})]_{0} \cong \begin{cases} K^{\alpha -1} & \text{if } \;i=1 \\ 0 & \text{if } \;i \neq 1 , d \end{cases} \] where \(d = \dim(S/I_{\Delta}) = \dim(\Delta) + 1\) and \(\alpha\) is the number of connected components of \(\Delta\). Let \(K[\Delta]\) be a Buchsbaum Stanley-Reisner ring with \(\dim(K[\Delta]) = d+1\) and core\((\Delta) = \Delta\). The following conditions are equivalent: (1) \(K[\Delta]\) has minimal multiplicity in the sense of \textit{S. Goto} [J. Algebra 85, 490--534 (1983; Zbl 0529.13010)]; (2) \(\Delta\) is a disjoint union of \(d\)-simplices; (3) \(I_{\Delta}\) is matroidal of degree \(2\). For the Buchsbaum Stanley-Reisner ring \(K[\Delta]\), if one of above conditions holds then the power of the Stanley-Reisner ideal \(I_{\Delta}^{\ell}\) has finite local cohomology and has a linear resolution for all \(\ell \geq 1\).