| On the Depth and Hilbert Series of the Fiber Cone |
| Received:February 12, 2008 Revised:January 05, 2009 |
| Key Words:
Cohen-Macaulay local ring fiber cone depth Hilbert series associated graded ring multiplicity.
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10771152). |
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| Abstract: |
| 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 $G(I)\ge d-1$. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2010.02.021 |
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