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Let $R$ be a noetherian ring and $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$- module. I know two fact. first, dimension of $M$(i.e. krull dimension of $R/{\rm ann}(M)$) is greater than or equal to cohomological dimension of $M$ with respect to $I$. and second, arithmetic rank of $I$(i.e. ${\rm inf}\{r\in \mathbb{N}_0 | \exists x_1, \cdots ,x_r \in R~\mbox{such that}~ \sqrt{<x_1, \cdots,x_r>}=\sqrt{I}\}$) is greater then or equal to cohomological dimension of $M$ with respect to $I$.

I wonder when equalty holds...

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  • $\begingroup$ 1) Your question needs an edit: you need $\sqrt{(x_1,\ldots,x_r)}=\sqrt{I}$, but for some reason this is not showing in your question properly. 2) You want which equality to hold? Do you want all three quantities (dimension, cohomological dimension and arithmetic rank) to be equal? Or do you want dimension and arithmetic rank to be equal? Or do you want dimension to be equal to cohomological dimension? etc. $\endgroup$
    – Mahdi Majidi-Zolbanin
    Commented Jan 2, 2013 at 15:55
  • $\begingroup$ @Mahdi Majidi-Zolbanin 1)I need each case equality holds. 2) example for strictly greater Thanks. $\endgroup$
    – Lee sangcheol
    Commented Jan 3, 2013 at 15:08

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