Herzog, Jürgen; Zheng, Xinxian Bounds for Hilbert coefficients. (English) Zbl 1162.13010 Proc. Am. Math. Soc. 137, No. 2, 487-494 (2009). Let \(K\) be a field, \(S =K[x_{1}, \ldots , x_{n}]\) the polynomial ring in \(n\) variables, and let \(N\) be any graded \(S\)-module of dimension \(d\). If \(N\) has a pure resolution, the authors give explicit formulas for the Hilbert coefficients \(e_{i}(N)\). Moreover, the combination of this result and some recent work of D. Eisenbud and F.-O. Schreyer [“Betti numbers of graded modules and cohomology of vector bundles”, arXiv:0712.1843, Theorem 0.2] is able to give lower and upper bounds for these coefficients \(e_{i}(N)\) for arbitrary graded modules \(N\). Reviewer: Najib Mahdou (Fès) Cited in 2 ReviewsCited in 4 Documents MSC: 13H15 Multiplicity theory and related topics 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13D02 Syzygies, resolutions, complexes and commutative rings Keywords:Hilbert coefficients; pure resolutions; multiplicity Software:BoijSoederberg PDFBibTeX XMLCite \textit{J. Herzog} and \textit{X. Zheng}, Proc. Am. Math. Soc. 137, No. 2, 487--494 (2009; Zbl 1162.13010) Full Text: DOI arXiv References: [1] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. · Zbl 0788.13005 [2] M. Boij and J. Söderberg, Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture, arXiv:math/0612047. · Zbl 1259.13009 [3] D. Eisenbud and F. Schreyer, Betti numbers of graded modules and cohomology of vector bundles, arXiv:0712.1843. New Version: Preprint, 2008. [4] Christopher A. Francisco and Hema Srinivasan, Multiplicity conjectures, Syzygies and Hilbert functions, Lect. Notes Pure Appl. Math., vol. 254, Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 145 – 178. · Zbl 1125.13011 · doi:10.1201/9781420050912.ch5 [5] J. Herzog and M. Kühl, On the Betti numbers of finite pure and linear resolutions, Comm. Algebra 12 (1984), no. 13-14, 1627 – 1646. · Zbl 0543.13008 · doi:10.1080/00927878408823070 [6] Jürgen Herzog and Hema Srinivasan, Bounds for multiplicities, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2879 – 2902. · Zbl 0899.13026 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.