Question

Let $(R, m)$ be a Noetherian local ring with infinite residue field and $I$ an ideal. If $\mathrm{r}(I)=s$, show that $\mathrm{r}\left(I^q\right) \leq \ell(I)$ for all $q \geq s$.

   Let $(R, m)$ be a Noetherian local ring with infinite residue field and $I$ an ideal. If $\mathrm{r}(I)=s$, show that $\mathrm{r}\left(I^q\right) \leq \ell(I)$ for all $q \geq s$.

Integral Closure: Rees Algebras, Multiplicities, Algorithms

Integral Closure: Rees Algebras, Multiplicities, Algorithms

Wolmer Vasconcelos 1st Edition

Chapter 1, Problem 9

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Step 1

In this problem, $r(I)$ represents the reduction number of the ideal $I$, and $\ell(I)$ represents the analytic spread of $I$. The reduction number $r(I)$ is the smallest integer $n$ such that $I^{n+1} = JI^n$ for some minimal reduction $J$ of $I$. The analytic Show more…

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Let $(R, m)$ be a Noetherian local ring with infinite residue field and $I$ an ideal. If $\mathrm{r}(I)=s$, show that $\mathrm{r}\left(I^q\right) \leq \ell(I)$ for all $q \geq s$.

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