Candidates for nonzero Betti numbers of monomial ideals. Issue 4 (3rd April 2017)

• Record Type: Journal Article
• Authors: Yazdan Pour, Ali Akbar
• Is Part Of: Communications in algebra. Volume 45:Issue 4(2017)
• Page Start: 1483
• Page End: 1492
• Publication Date: 2017-04-03
• Subjects: Betti numbers -- Fröberg's theorem -- Green–Lazarsfel index -- syzygy; Primary 13D02 -- 13D05 -- Secondary 05E40; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2016.1177828
• Languages: English
• ISSNs: 0092-7872

Abstract

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 elements of degrees j, …, j +( d −1) (where j ≥ i + d ), then ( i +1)-th 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 2 with linear resolution. Among all, we deduce a partial result on subadditivity of the syzygies for monomial ideals.