Candidates for nonzero Betti numbers of monomial ideals

DOI10.1080/00927872.2016.1177828zbMATH Open1370.13015arXiv1507.07188OpenAlexW2964250889MaRDI QIDQ2984976

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Publication date: 12 May 2017

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Abstract: 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 $j,ldots,j+(d-1)$ (where $jgeqi+d$), 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 Fr"oberg'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 $$.

Full work available at URL: https://arxiv.org/abs/1507.07188

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