Buchsbaum and Eulerian complexes
Let \(\Delta\) denote a finite simplicial complex. For a fixed field \(K\) let \(K [\Delta]\) denote its Stanley-Reisner ring over \(K\). Then \(\Delta\) is called a Cohen-Macaulay (resp. Buchsbaum) complex over \(K\) if and only if \(K [\Delta]\) is a Cohen-Macaulay (resp. Buchsbaum) ring. This is characterized topologically by the vanishing of the simplicial cohomology of the links of \(\Delta\) with coefficients in \(K\) [see \textit{G. A. Reisner}, Adv. Math. 21, 30-49 (1976; Zbl 0345.13017); respectively the reviewer, Math. Z. 178, 125-142 (1981; Zbl 0472.13012)]. For a Buchsbaum complex \(\Delta\) let \(\beta (\Delta) : = (\beta_0, \dots, \beta_{d - 1})\) with \(\beta_i : = \dim_KH_i (\Delta,K)\) and \(0 \leq i \leq d - 1 : = \dim \Delta\). Then \(\chi (\Delta) = \sum^{d - 1}_{i = 0} (-1)^i \beta_i (\Delta)\) is the Euler characteristic of \(\Delta\). Moreover \(\Delta\) is called an Eulerian complex provided it is pure and for every face \(F \in \Delta\) one has \(\chi (\text{lk}_\Delta F) = 1 + (-1)^{d(F)}\), where \(d(F)\) denotes the dimension of the link \(\text{lk}_\Delta F\). Then the authors describe the possible \(\beta (\Delta)\)'s for Buchsbaum-Eulerian, Eulerian, and semi-Eulerian complexes. Furthermore, they describe the ring-theoretic Betti numbers of \(K [\Delta]\) in a free resolution, in particular the depth of \(K [\Delta]\), in the case of Buchsbaum-Eulerian complexes. It turns out that for an Eulerian complex any depth between one and the dimension is possible.
scientific article; zbMATH DE number 737476
scientific article