On the depth of the tangent cone and the growth of the Hilbert function

Elias, Juan

doi:10.1090/S0002-9947-99-02278-3

On the depth of the tangent cone
and the growth of the Hilbert function

Author: Juan Elias
Journal: Trans. Amer. Math. Soc. 351 (1999), 4027-4042
MSC (1991): Primary 13A30, 13D40, 13H10
DOI: https://doi.org/10.1090/S0002-9947-99-02278-3
Published electronically: April 20, 1999
MathSciNet review: 1491860
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Abstract: For a dimensional Cohen-Macaulay local ring $(R, \mathbf{m})$ we study the depth of the associated graded ring of with respect to an $\textbf{ m}$ -primary ideal in terms of the Vallabrega-Valla conditions and the length of $I^{t+1}/JI^{t}$ , where is a minimal reduction of and $t\ge 1$ . As a corollary we generalize Sally's conjecture on the depth of the associated graded ring with respect to a maximal ideal to $\mathbf{m}$ -primary ideals. We also study the growth of the Hilbert function.

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