Bounds for the Betti numbers of successive stellar subdivisions of a simplex. (English) Zbl 1360.13053

Consider the class of simplicial complexes obtained from the boundary complex of a simplex with \(q + 1\) vertices by any sequence of \(c-i\) stellar subdivisions. There are bounds for the (total) Betti numbers of the minimal resolution of various classes of simplicial complexes. If we only subdivide facets starting from a simplex the process will yield a stacked polytope. In this case, there is an explicit formula for the Betti numbers due to N. Terai and T. Hibi [Manuscr. Math. 92, No. 4, 447–453 (1997; Zbl 0882.13018)]. See also S. Choi and J. S. Kim [Electron. J. Comb. 17, No. 1, Research Paper R9, 8 p. (2010; Zbl 1215.05011)] for a combinatorial proof, and J. Herzog and E. M. Li Marzi [Lect. Notes Pure Appl. Math. 206, 157–167 (1999; Zbl 0941.52013)] for the costruction of the resolutions.
For \(c \geq 1\) and \(q\geq 2\), \(\mathcal{D}_{q,c}\) is the set of simplicial subcomplexes \(D\subset 2^{[q+c]}\) such that there exists a sequence of simplicial complexes \[ D_1, D_2,\cdots , D_{c-1}, D_c = D \] with the property that \(D_1\) is the boundary complex of the simplex on \(q + 1\) vertices, and, for \(0\leq i\leq c-1\), \(D_{i+1}\subset 2^{[q+i]}\) is obtained from \(D_i \subset 2^{[q+i]}\) by a stellar subdivision of a face of \(D_i\) of dimension at least 1 with new vertex \(q + i + 1\). \(\mathrm{supp }D_i = [q + i]\) and \(\mathrm{codim }k[D_i] = i\) for all \(i\). Assume \(D \in {\mathcal{D}}_{q,c}\) and \(k\) be any field. Consider the Stanley-Reisner ring \(k[D] =R_{[q+c]}/I_D\). Note that \(k[D]\) is the quotient of a polynomial ring by a Gorenstein ideal with codimension \(c\). Define inductively \(l_c = (l_{c,0}, l_{c,1},\cdots,l_{c,c})\in \mathbb{Z}^{c+1}\) by \(l_1 = (1, 1)\) and \[ l_c = 2(l_{c-1}, 0) + 2(0, l_{c-1})- (1,1,0,\cdots,0)-(0,\cdots,0,1,1)\in\mathbb{Z}^{c+1} \] for \(c\geq 2\). The paper under review gives the bound for the Betti numbers of the Stanley-Reisner ring of a stellar subdivision of a Gorenstein* simplicial complex. The authors prove that \(b_i(k[D]) \leq l_{c,i}\) for all \(0\leq i\leq c\). Their main tool is the relation of stellar subdivisions of Gorenstein* simplicial complexes with the Kustin-Miller complex construction obtained in [J. Böhm and S. A. Papadakis, Australas. J. Comb. 55, 235–247 (2013; Zbl 1282.13041)]. They show that the bound depends only on the number of subdivisions, and they construct examples which prove that it is sharp.


13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13D02 Syzygies, resolutions, complexes and commutative rings
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