# Cohomological dimension, dimension of modules and arithmetic rank

Let $R$ be a noetherian ring, $I$ an ideal of $R$ and $M$ a finitely generated $R$- module.
I know two facts: 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. $\inf\{r\in \mathbb{N}_0 \mid \exists x_1, \cdots ,x_r \in R~\mbox{such that}~ \sqrt{\langle x_1, \dots,x_r\rangle}=\sqrt{I}\}$) is greater then or equal to cohomological dimension of $M$ with respect to $I$.

I wonder when equality holds...

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