2015/07/26 by Ali Akbar Yazdan Pour, Pour, Ali Akbar Yazdan
Mathematics · #13D05 #Algebraic Topology (math.AT) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13D02 #Secondary 05E40 #math.AC #math.AT #msc:05E40 #msc:13D02 #msc:13D05
paper · pdf · doi:10.48550/arxiv.1507.07188
arxiv created 2016/07/03 · arxiv updated 2016/07/05
Let I be a monomial ideal in the polynomial ring S generated by elements of degree at most d. In this paper, it is shown that, if the i-th syzygy of I has no element of degrees j, …, j+(d-1) (where j ≥ i+d), then (i+1)-syzygy of I does not have any element of degree j+d. Then we give several applications of this result, including an alternative proof for Green-Lazarsfeld index of the edge ideals of graphs as well as an alternative proof for Fröberg's theorem on classification of square-free monomial ideals generated in degree two with linear resolution. Among all, we describe the possible indices i, j for which I may have non-zero Betti numbers βi,j.