Linear syzygies, flag complexes, and regularity

Abstract

We show that for every \(r\in \mathbb {Z}_{>0}\) there exist monomial ideals generated in degree two, with linear syzygies, and regularity of the quotient equal to \(r\). For Gorenstein ideals we prove that the regularity of their quotients can not exceed four, thus showing that for \(d > 4\) every triangulation of a \(d\)-manifold has a hollow square or simplex. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity is \(O(\log (\log (n))\), where \(n\) is the number of variables.