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A $d$-dimensional extension of a lemma of Huneke's and formulas for the Hilbert coefficients

Author: Sam Huckaba
Journal: Proc. Amer. Math. Soc. 124 (1996), 1393-1401
MSC (1991): Primary 13D40, 13A30, 13H10
MathSciNet review: 1307529
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Abstract: A $d$-dimensional version is given of a $2$-dimensional result due to C. Huneke. His result produced a formula relating the length $\lambda (I^{n+1}/JI^{n})$ to the difference $P_{I}(n+1)-H_{I}(n+1)$, where $I$ is primary for the maximal ideal of a $2$-dimensional Cohen-Macaulay local ring $R$, $J$ is a minimal reduction of $I$, $H_{I}(n)=\lambda (R/I^{n})$, and $P_{I}(n)$ is the Hilbert-Samuel polynomial of $I$. We produce a formula that is valid for arbitrary dimension, and then use it to establish some formulas for the Hilbert coefficients of $I$. We also include a characterization, in terms of the Hilbert coefficients of $I$, of the condition $depth(G(I))\geq d-1$.

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Additional Information

Sam Huckaba
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306-3027

Keywords: Hilbert-Samuel polynomial, depth, associated graded ring, Cohen-Macaulay
Received by editor(s): June 8, 1994
Received by editor(s) in revised form: November 8, 1994
Additional Notes: The author was partially supported by the NSA (#MDA904-92-H-3040).
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1996 American Mathematical Society

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