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Bounds for Hilbert coefficients. (English) Zbl 1162.13010

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 \(N\).

MSC:

13H15 Multiplicity theory and related topics
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13D02 Syzygies, resolutions, complexes and commutative rings

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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
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