Rational powers of ideals in Noetherian rings

Question

Let $R$ be a Noetherian ring, and let $I$ be an ideal of $R$. Fix a rational number $a=\frac{p}{q}$ with $p, q\in \mathbb{Z_\geq 0}$ $q\neq 0$. We define $I_a = \{x \in R: x^q\in \overline{I^p}\}$, where overline denotes integral closure.

Huneke-Swanson show, (in their book on Integral Closure), that $I_a$ is well defined, is indeed an ideal and is integrally closed. If $a\leq b$, $I_b\subseteq I_a$ and if $n$ is a positive integer, then, $I_n=\overline{I^n}$.

I was curious to know what else is known about these rational powers. Are there any computations done in special cases (e.g. monomial ideals)? Do these ideals find any use other than computing the integral closure of the extended Rees ring in the Laurent polynomial ring? I would appreciate if anyone has any references on this subject other than Huneke-Swanson book.

So far I have only found a paper by David Rush "Rees valuations and asymptotic primes of rational powers in Noetherian rings and lattices"

PS: I am not sure why, but I am unable to typeset curly braces (I am using backslash followed by brace sign, but it does not show up) and the "less than" (causes text after the symbol to vanish) symbol.

Answer 1

I give some hints about computing them in my paper Balanced normal cones and Fulton-MacPherson's intersection theory, section 3. (Also I compute 14 examples.)

I find it kind of astounding that this easy and beautiful construction has sat essentially unused in the literature for over 50 years now. There is Rees' book on asymptotic properties of ideals, and the Huneke-Swanson book, but that's almost the entire literature.

Answer 2

This isn't a real answer, but perhaps a direction to look for similar ideas:

You define $$ I_a = \{ x \in R | x^q \in {\overline{I^p}} \}. $$ Suppose $R$ is a (normal?) domain, set $\pi : Y \to \text{Spec} R$ to be the normalized blow-up of $\text{Spec} R$ at $I$ and set $I \mathcal{O}_Y = \mathcal{O}_Y(-G)$. Now, rephrasing above, we have that $$ I_a = \{ x \in R | q \text{Div}_Y(x) \geq p G \}. $$ The inequality in the display can also be phrased as $Div_Y(x) \geq a G$ ($aG$ is a $\mathbb{Q}$-divisor), or in other words that $Div_Y(x) \geq \lceil a G \rceil$ (here rounding up rounds each divisor coefficient separately). Thus, this is the same as $x \in \pi_* \mathcal{O}_Y(-\lceil a G \rceil)$.

This looks similar (although is obviously different) than the multiplier ideal of $(R, I^a)$ which, in the case that $R$ is ($\mathbb{Q}$)-Gorenstein (for example, regular), is defined to be: $$ \pi_* O_Y(\lceil K_Y - \pi^* K_{X} - a G \rceil). $$ In the regular case, $K_Y - \pi^*K_X$ is just the Jacobian of $\pi$.