2021/05/25 by Ralf Fröberg, Fröberg, Ralf · 1 citation
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Commutative Algebra and Its Applications #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.2105.12025
Let k be a field and R=k[x1,…,xn]/I=S/I a graded ring. Then R has a t-linear resolution if I is generated by homogeneous elements of degree t, and all higher syzygies are linear. Thus R has a t-linear resolution if \rm TorSi,j(S/I,k)=0 if j≠ i+t-1. For a simplicial complex Δ on [\bf n]=\1,…,n\ and a field k, the Stanley-Reisner ring k[Δ] is k[x1,…,xn]/I, where I is generated by those squarefree monomials xi1⋯ xik for which \ i1,…,ik\ does not belong to Δ. In \citeFr the Stanley-Reisner rings with 2-linear resolution are determined. Their associated complexes has had different names in the literature. We call them fat forests here. In this article we determine the Betti numbers of fat forests. We also consider Betti numbers of Alexander duals of fat forests.