Betti numbers of fat forests and their Alexander dual

Abstract

AbstractLet k be a field and $$R=k[x_1,\ldots ,x_n]/I=S/I$$ R = k [ x 1 , … , x n ] / 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 $$\mathrm{Tor}^S_{i,j}(S/I,k)=0$$ Tor i , j S ( S / I , k ) = 0 if $$j\ne i+t-1$$ j ≠ i + t - 1 . For a graph G on $$\{1,\ldots ,n\}$$ { 1 , … , n } , the edge algebra is $$k[x_1,\ldots ,x_n]/I$$ k [ x 1 , … , x n ] / I , where I is generated by those $$x_ix_j$$ x i x j for which $$\{ i,j\}$$ { i , j } is an edge in G. We want to determine the Betti numbers of edge rings with 2-linear resolution. But we want to do that by looking at the edge ring as a Stanley–Reisner ring. For a simplicial complex $$\Delta $$ Δ on $$[\mathbf{n}]=\{1,\ldots ,n\}$$ [ n ] = { 1 , … , n } and a field k, the Stanley–Reisner ring $$k[\Delta ]$$ k [ Δ ] is $$k[x_1,\ldots ,x_n]/I$$ k [ x 1 , … , x n ] / I , where I is generated by the squarefree monomials $$x_{i_1}\ldots x_{i_k}$$ x i 1 … x i k for which $$\{ i_1,\ldots ,i_k\}$$ { i 1 , … , i k } does not belong to $$\Delta $$ Δ . Which Stanley–Reisner rings that are edge rings with 2-linear resolution are known. Their associated complexes has had different names in the literature. We call them fat forests here. We determine the Betti numbers of many fat forests and compare our result with what is known. We also consider Betti numbers of Alexander duals of fat forests.