First Nonlinear Syzygies of Ideals Associated to Graphs

2008/11/12 by Oscar Fernandez-Ramos, Fernandez-Ramos, Oscar, Philippe Gimenez +1
Mathematics · #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.CO

paper · pdf · doi:10.48550/arxiv.0811.1865

11 pages

arxiv created 2008/11/12 · arxiv updated 2009/12/01

Abstract

Consider an ideal I⊂ K[x₁,..., 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 β_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.

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