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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hilbert-Samuel functions of Cohen-Macaulay rings
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by M. Boratyński and J. Święcicka PDF
Proc. Amer. Math. Soc. 51 (1975), 19-24 Request permission

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
  • Eben Matlis, The multiplicity and reduction number of a one-dimensional local ring, Proc. London Math. Soc. (3) 26 (1973), 273–288. MR 313247, DOI 10.1112/plms/s3-26.2.273
  • P. Samuel, La notion de multiplicité en algèbre et en géométrie algébrique, Thèse, Gauthier-Villars, Paris, 1951. 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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 51 (1975), 19-24
  • MSC: Primary 13H10; Secondary 13H15
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0432630-0
  • MathSciNet review: 0432630