Graded Betti numbers of balanced simplicial complexes

2018/11/09 by Juhnke-Kubitzke, Martina, Venturello, Lorenzo
#05E40 #05E45 #13F55 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1811.03892

Abstract

We prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings S/I, where S is a polynomial ring and I⊆ S is an homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.

Related

• The Graded Betti Numbers of the Skeletons of Simplicial Complexes
• Homological invariants of the Stanley-Reisner ring of a k-decomposable simplicial complex
• Betti numbers of Stanley--Reisner rings with pure resolutions
• Betti numbers of graded modules and the Multiplicity Conjecture in the non-Cohen-Macaulay case
• Betti numbers of skeletons