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Hilbert-Samuel functions of Cohen-Macaulay rings

Authors: M. Boratyński and J. Święcicka
Journal: Proc. Amer. Math. Soc. 51 (1975), 19-24
MSC: Primary 13H10; Secondary 13H15
MathSciNet review: 0432630
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Abstract: Let $ R$ be a local ring with a maximal ideal $ \mathfrak{m}$. It is proved that in case $ R$ is a Cohen-Macaulay (C.M.) ring and $ \dim \mathfrak{m}/{\mathfrak{m}^2} - \dim R = 1$, then the multiplicity of $ R$ and its dimension determine uniquely the Hilbert-Samuel function of $ R$. As a corollary we obtain that the C.M. property is determined by the Hilbert-Samuel function in case $ \dim \mathfrak{m}/{\mathfrak{m}^2} - \dim R = 1$. An example is given which shows that it is not so in case $ \dim \mathfrak{m}/{\mathfrak{m}^2} - \dim R > 1$.

References [Enhancements On Off] (What's this?)

  • [1] Eben Matlis, The multiplicity and reduction number of a one-dimensional local ring, Proc. London Math. Soc. (3) 26 (1973), 273–288. MR 0313247,
  • [2] P. Samuel, La notion de multiplicité en algèbre et en géométrie algébrique, Thèse, Gauthier-Villars, Paris, 1951.
  • [3] J.-P. Serre, Algèbre locale. Multiplicités, 2nd rev. ed., Lecture Notes in Math., no. 11, Springer-Verlag, Berlin and New York, 1965. MR 34 #1352.

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Keywords: Cohen-Macaulay ring, Hilbert-Samuel function
Article copyright: © Copyright 1975 American Mathematical Society