Question

Let $I$ be an ideal generated by a regular sequence of $n$ elements. Show that for every positive integer $s$, $$ \mathrm{r}\left(I^s\right)=\left\lceil\frac{(n-1)(s-1)}{s}\right\rceil $$

   Let $I$ be an ideal generated by a regular sequence of $n$ elements. Show that for every positive integer $s$,

$$
\mathrm{r}\left(I^s\right)=\left\lceil\frac{(n-1)(s-1)}{s}\right\rceil
$$

Integral Closure: Rees Algebras, Multiplicities, Algorithms

Integral Closure: Rees Algebras, Multiplicities, Algorithms

Wolmer Vasconcelos 1st Edition

Chapter 1, Problem 18

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We have an ideal $I$ generated by a regular sequence of $n$ elements, and we need to show that the regularity of $I^s$ is given by $r(I^s) = \lceil\frac{(n-1)(s-1)}{s}\rceil$ for any positive integer $s$. Show more…

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Let $I$ be an ideal generated by a regular sequence of $n$ elements. Show that for every positive integer $s$, $$ \mathrm{r}\left(I^s\right)=\left\lceil\frac{(n-1)(s-1)}{s}\right\rceil $$

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