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# The Hilbert Coefficients of the Fiber Cone and the $a$-Invariant of the Associated Graded Ring

Published:2009-08-01
Printed: Aug 2009
• Clare D'Cruz
• Tony J. Puthenpurakal
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## Abstract

Let $(A,\m)$ be a Noetherian local ring with infinite residue field and let $I$ be an ideal in $A$ and let $F(I) = \bigoplus_{n \geq 0}I^n/\m I^n$ be the fiber cone of $I$. We prove certain relations among the Hilbert coefficients $f_0(I),f_1(I), f_2(I)$ of $F(I)$ when the $a$-invariant of the associated graded ring $G(I)$ is negative.
 Keywords: fiber cone, $a$-invariant, Hilbert coefficients of fiber cone
 MSC Classifications: 13A30 - Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13D40 - Hilbert-Samuel and Hilbert-Kunz functions; Poincare series