Ralf Fröberg: On squarefree monomial rings with 2-linear resolution

Abstract:

For a graph \(G\) with vertices \(V=\{v_1,\ldots,v_n\}\) and edges \(E\), the edge ring \(k[G]\) is \(k[v_1,\ldots,v_n]/I\), where \(I\) is generated by all \(v_iv_j\) for which \((i,j)\in E\). For which graphs does \(k[G]\) have a linear resolution? Which Stanley-Reisner rings have 2-linear resolution? That a ring \(k[x_1,\ldots,x_n]/I=S/I\) has 2-linear resolution means that \(I\) is generated in degree 2 and has a linear resolution, equivalently that Tor\(_i^S(I,k)_j=0\) if \(j\ne i+2\). What are the Betti numbers of the skeletons of those complexes?