On the Depth and Hilbert Series of the Fiber Cone
Received:February 12, 2008  Revised:January 05, 2009
Key Word: Cohen-Macaulay local ring   fiber cone   depth   Hilbert series   associated graded ring   multiplicity.
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.10771152).
 Author Name Affiliation Guang Jun ZHU School of Mathematical Science, Suzhou University, Jiangsu 215006, P. R. China
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Let $(R,\frak{m})$ be a Cohen-Macaulay local ring of dimension $d$ with infinite residue field, $I$ an $\frak{m}$-primary ideal and $K$ an ideal containing $I$. Let $J$ be a minimal reduction of $I$ such that, for some positive integer $k$, $KI^n\cap J=JKI^{n-1}$ for $n\le k-1$ and $\lambda(\frac{KI^{k}}{JKI^{k-1}})=1$. We show that if depth $G(I)\ge d-2$, then such fiber cones have almost maximal depth. We also compute, in this case, the Hilbert series of $F_K(I)$ assuming that depth $G(I)\ge d-1$.