FIRST NONLINEAR SYZYGIES OF IDEALS ASSOCIATED TO GRAPHS

Consider an ideal I subset of K[x(1),..., x(n)], with K an arbitrary field, generated by monomials of degree two. Assuming that I does not have a linear resolution, we determine the step s of the minimal graded free resolution of I where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree s + 3, and we compute the corresponding graded Betti number beta(s,s+3). The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1.