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Syzygies of the Veronese Modules
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2016 (English)In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 44, no 9, 3890-3906 p.Article in journal (Refereed) PublishedText
Abstract [en]

We study the minimal free resolution of the Veronese modules, S-n,S- d,S- k = circle plus S-i >= 0(k+id), where S = K[x(1), ... , x(n)], by giving a formula for the Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. We prove that S-n,S- d,S- k is Cohen-Macaulay if and only if k < d, and that its minimal resolution is pure and has some linearity features when k > d (n - 1) - n. We prove combinatorially that the resolution of S-2,S- d,S- k is pure. We show that HS(S-n,S- d,S- k; z) = 1/(n-1)d(n-1)/dz(n-1) [z(k+n-1)/1-z(d)]. As an application, we calculate the complete Betti diagrams of the Veronese rings K[x, y, z]((d)), for d = 4, 5, and K[x, y, z, u]((3)).

Place, publisher, year, edition, pages
Taylor & Francis, 2016. Vol. 44, no 9, 3890-3906 p.
Keyword [en]
Betti Numbers, Hilbert Series, Pile simplicial complex, Veronese Modules
National Category
URN: urn:nbn:se:kth:diva-189102DOI: 10.1080/00927872.2015.1027389ISI: 000377029600014OAI: diva2:943800

QC 20160628

Available from: 2016-06-28 Created: 2016-06-27 Last updated: 2016-06-28Bibliographically approved

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