On the Upper bound of the Multiplicity Conjecture

Details On the Upper bound of the Multiplicity Conjecture On the Upper bound of the Multiplicity Conjecture

2007-01-27

arxiv.org/abs/math/0701793v2

Mathematics

Commutative Algebra

6 pages, Many typos corrected. An additional section on examples added. To appear in Proc. of AMS

Scientific paper

Let $A = K[X_1,...,X_n]$ and let $I$ be a graded ideal in $A$. We show that the upper bound of Multiplicity conjecture of Herzog, Huneke and Srinivasan holds asymptotically (i.e., for $I^k$ and all $k \gg 0$) if $I$ belongs to any of the following large classes of ideals: \begin{enumerate}[\rm (1)] \item radical ideals. \item monomial ideals with generators in different degrees. \item zero-dimensional ideals with generators in different degrees. \end{enumerate} Surprisingly, our proof uses local techniques like analyticity, reductions, equimultiplicity and local results like Rees's theorem on multiplicities.

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